0.10/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.10/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n014.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 13:51:24 EDT 2019
0.12/0.33	% CPUTime    : 
72.79/73.06	% SZS status Unsatisfiable
72.79/73.06	
72.86/73.07	% SZS output start Proof
72.86/73.07	Take the following subset of the input axioms:
72.87/73.07	  fof(composition_associativity_5, axiom, ![A, C, B]: composition(composition(A, B), C)=composition(A, composition(B, C))).
72.87/73.07	  fof(composition_distributivity_7, axiom, ![A, C, B]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
72.87/73.07	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
72.87/73.07	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
72.87/73.07	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
72.87/73.07	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
72.87/73.07	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
72.87/73.07	  fof(dedekind_law_14, axiom, ![A, C, B]: composition(meet(A, composition(C, converse(B))), meet(B, composition(converse(A), C)))=join(meet(composition(A, B), C), composition(meet(A, composition(C, converse(B))), meet(B, composition(converse(A), C))))).
72.87/73.07	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
72.87/73.07	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
72.87/73.07	  fof(goals_17, negated_conjecture, composition(sk1, top)=sk1).
72.87/73.07	  fof(goals_18, negated_conjecture, composition(meet(sk1, one), sk2)!=join(meet(sk1, sk2), composition(meet(sk1, one), sk2))).
72.87/73.07	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
72.87/73.07	  fof(maddux2_join_associativity_2, axiom, ![A, C, B]: join(join(A, B), C)=join(A, join(B, C))).
72.87/73.07	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
72.87/73.07	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
72.87/73.07	  fof(modular_law_1_15, axiom, ![A, C, B]: join(meet(composition(A, B), C), meet(composition(A, meet(B, composition(converse(A), C))), C))=meet(composition(A, meet(B, composition(converse(A), C))), C)).
72.87/73.07	  fof(modular_law_2_16, axiom, ![A, C, B]: meet(composition(meet(A, composition(C, converse(B))), B), C)=join(meet(composition(A, B), C), meet(composition(meet(A, composition(C, converse(B))), B), C))).
72.87/73.07	
72.87/73.07	Now clausify the problem and encode Horn clauses using encoding 3 of
72.87/73.07	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
72.87/73.07	We repeatedly replace C & s=t => u=v by the two clauses:
72.87/73.07	  fresh(y, y, x1...xn) = u
72.87/73.07	  C => fresh(s, t, x1...xn) = v
72.87/73.07	where fresh is a fresh function symbol and x1..xn are the free
72.87/73.07	variables of u and v.
72.87/73.07	A predicate p(X) is encoded as p(X)=true (this is sound, because the
72.87/73.07	input problem has no model of domain size 1).
72.87/73.07	
72.87/73.07	The encoding turns the above axioms into the following unit equations and goals:
72.87/73.07	
72.87/73.07	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
72.87/73.07	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
72.87/73.07	Axiom 3 (composition_identity_6): X = composition(X, one).
72.87/73.07	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
72.87/73.07	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
72.87/73.07	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
72.87/73.07	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
72.87/73.07	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
72.87/73.07	Axiom 9 (def_top_12): join(X, complement(X)) = top.
72.87/73.07	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
72.87/73.07	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
72.87/73.07	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
72.87/73.07	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
72.87/73.07	Axiom 14 (modular_law_2_16): meet(composition(meet(X, composition(Y, converse(Z))), Z), Y) = join(meet(composition(X, Z), Y), meet(composition(meet(X, composition(Y, converse(Z))), Z), Y)).
72.87/73.07	Axiom 15 (modular_law_1_15): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
72.87/73.07	Axiom 16 (dedekind_law_14): composition(meet(X, composition(Y, converse(Z))), meet(Z, composition(converse(X), Y))) = join(meet(composition(X, Z), Y), composition(meet(X, composition(Y, converse(Z))), meet(Z, composition(converse(X), Y)))).
74.81/75.06	Axiom 17 (goals_17): composition(sk1, top) = sk1.
74.81/75.06	
74.81/75.06	Lemma 18: meet(X, Y) = meet(Y, X).
74.81/75.06	Proof:
74.81/75.06	  meet(X, Y)
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  complement(join(complement(X), complement(Y)))
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  complement(join(complement(Y), complement(X)))
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  meet(Y, X)
74.81/75.06	
74.81/75.06	Lemma 19: complement(top) = zero.
74.81/75.06	Proof:
74.81/75.06	  complement(top)
74.81/75.06	= { by axiom 9 (def_top_12) }
74.81/75.06	  complement(join(complement(?), complement(complement(?))))
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  meet(?, complement(?))
74.81/75.06	= { by axiom 8 (def_zero_13) }
74.81/75.06	  zero
74.81/75.06	
74.81/75.06	Lemma 20: complement(join(zero, complement(X))) = meet(X, top).
74.81/75.06	Proof:
74.81/75.06	  complement(join(zero, complement(X)))
74.81/75.06	= { by lemma 19 }
74.81/75.06	  complement(join(complement(top), complement(X)))
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  meet(top, X)
74.81/75.06	= { by lemma 18 }
74.81/75.06	  meet(X, top)
74.81/75.06	
74.81/75.06	Lemma 21: converse(composition(converse(X), Y)) = composition(converse(Y), X).
74.81/75.06	Proof:
74.81/75.06	  converse(composition(converse(X), Y))
74.81/75.06	= { by axiom 10 (converse_multiplicativity_10) }
74.81/75.06	  composition(converse(Y), converse(converse(X)))
74.81/75.06	= { by axiom 5 (converse_idempotence_8) }
74.81/75.06	  composition(converse(Y), X)
74.81/75.06	
74.81/75.06	Lemma 22: composition(converse(one), X) = X.
74.81/75.06	Proof:
74.81/75.06	  composition(converse(one), X)
74.81/75.06	= { by lemma 21 }
74.81/75.06	  converse(composition(converse(X), one))
74.81/75.06	= { by axiom 3 (composition_identity_6) }
74.81/75.06	  converse(converse(X))
74.81/75.06	= { by axiom 5 (converse_idempotence_8) }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 23: composition(one, X) = X.
74.81/75.06	Proof:
74.81/75.06	  composition(one, X)
74.81/75.06	= { by lemma 22 }
74.81/75.06	  composition(converse(one), composition(one, X))
74.81/75.06	= { by axiom 12 (composition_associativity_5) }
74.81/75.06	  composition(composition(converse(one), one), X)
74.81/75.06	= { by axiom 3 (composition_identity_6) }
74.81/75.06	  composition(converse(one), X)
74.81/75.06	= { by lemma 22 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 24: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y).
74.81/75.06	Proof:
74.81/75.06	  join(complement(Y), composition(converse(X), complement(composition(X, Y))))
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(composition(converse(X), complement(composition(X, Y))), complement(Y))
74.81/75.06	= { by axiom 7 (converse_cancellativity_11) }
74.81/75.06	  complement(Y)
74.81/75.06	
74.81/75.06	Lemma 25: join(complement(X), complement(X)) = complement(X).
74.81/75.06	Proof:
74.81/75.06	  join(complement(X), complement(X))
74.81/75.06	= { by lemma 22 }
74.81/75.06	  join(complement(X), composition(converse(one), complement(X)))
74.81/75.06	= { by lemma 23 }
74.81/75.06	  join(complement(X), composition(converse(one), complement(composition(one, X))))
74.81/75.06	= { by lemma 24 }
74.81/75.06	  complement(X)
74.81/75.06	
74.81/75.06	Lemma 26: join(X, join(Y, Z)) = join(Z, join(X, Y)).
74.81/75.06	Proof:
74.81/75.06	  join(X, join(Y, Z))
74.81/75.06	= { by axiom 6 (maddux2_join_associativity_2) }
74.81/75.06	  join(join(X, Y), Z)
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(Z, join(X, Y))
74.81/75.06	
74.81/75.06	Lemma 27: join(X, join(complement(X), Y)) = join(Y, top).
74.81/75.06	Proof:
74.81/75.06	  join(X, join(complement(X), Y))
74.81/75.06	= { by lemma 26 }
74.81/75.06	  join(complement(X), join(Y, X))
74.81/75.06	= { by lemma 26 }
74.81/75.06	  join(Y, join(X, complement(X)))
74.81/75.06	= { by axiom 9 (def_top_12) }
74.81/75.06	  join(Y, top)
74.81/75.06	
74.81/75.06	Lemma 28: join(X, join(Y, complement(X))) = join(Y, top).
74.81/75.06	Proof:
74.81/75.06	  join(X, join(Y, complement(X)))
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(X, join(complement(X), Y))
74.81/75.06	= { by axiom 6 (maddux2_join_associativity_2) }
74.81/75.06	  join(join(X, complement(X)), Y)
74.81/75.06	= { by axiom 9 (def_top_12) }
74.81/75.06	  join(top, Y)
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(Y, top)
74.81/75.06	
74.81/75.06	Lemma 29: join(top, complement(X)) = top.
74.81/75.06	Proof:
74.81/75.06	  join(top, complement(X))
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(complement(X), top)
74.81/75.06	= { by lemma 28 }
74.81/75.06	  join(X, join(complement(X), complement(X)))
74.81/75.06	= { by lemma 25 }
74.81/75.06	  join(X, complement(X))
74.81/75.06	= { by axiom 9 (def_top_12) }
74.81/75.06	  top
74.81/75.06	
74.81/75.06	Lemma 30: join(X, top) = top.
74.81/75.06	Proof:
74.81/75.06	  join(X, top)
74.81/75.06	= { by axiom 9 (def_top_12) }
74.81/75.06	  join(X, join(complement(X), complement(complement(X))))
74.81/75.06	= { by lemma 27 }
74.81/75.06	  join(complement(complement(X)), top)
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(top, complement(complement(X)))
74.81/75.06	= { by lemma 29 }
74.81/75.06	  top
74.81/75.06	
74.81/75.06	Lemma 31: join(meet(X, Y), complement(join(complement(X), Y))) = X.
74.81/75.06	Proof:
74.81/75.06	  join(meet(X, Y), complement(join(complement(X), Y)))
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
74.81/75.06	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 32: join(zero, meet(X, top)) = X.
74.81/75.06	Proof:
74.81/75.06	  join(zero, meet(X, top))
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(meet(X, top), zero)
74.81/75.06	= { by lemma 19 }
74.81/75.06	  join(meet(X, top), complement(top))
74.81/75.06	= { by lemma 30 }
74.81/75.06	  join(meet(X, top), complement(join(complement(X), top)))
74.81/75.06	= { by lemma 31 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 33: join(meet(X, Y), meet(X, complement(Y))) = X.
74.81/75.06	Proof:
74.81/75.06	  join(meet(X, Y), meet(X, complement(Y)))
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(meet(X, complement(Y)), meet(X, Y))
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
74.81/75.06	= { by lemma 31 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 34: meet(X, zero) = zero.
74.81/75.06	Proof:
74.81/75.06	  meet(X, zero)
74.81/75.06	= { by lemma 18 }
74.81/75.06	  meet(zero, X)
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  complement(join(complement(zero), complement(X)))
74.81/75.06	= { by lemma 19 }
74.81/75.06	  complement(join(complement(complement(top)), complement(X)))
74.81/75.06	= { by lemma 25 }
74.81/75.06	  complement(join(complement(join(complement(top), complement(top))), complement(X)))
74.81/75.06	= { by lemma 19 }
74.81/75.06	  complement(join(complement(join(zero, complement(top))), complement(X)))
74.81/75.06	= { by lemma 20 }
74.81/75.06	  complement(join(meet(top, top), complement(X)))
74.81/75.06	= { by lemma 32 }
74.81/75.06	  complement(join(join(zero, meet(meet(top, top), top)), complement(X)))
74.81/75.06	= { by axiom 8 (def_zero_13) }
74.81/75.06	  complement(join(join(meet(top, complement(top)), meet(meet(top, top), top)), complement(X)))
74.81/75.06	= { by lemma 18 }
74.81/75.06	  complement(join(join(meet(top, complement(top)), meet(top, meet(top, top))), complement(X)))
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  complement(join(join(meet(top, complement(top)), meet(top, complement(join(complement(top), complement(top))))), complement(X)))
74.81/75.06	= { by lemma 25 }
74.81/75.06	  complement(join(join(meet(top, complement(top)), meet(top, complement(complement(top)))), complement(X)))
74.81/75.06	= { by lemma 33 }
74.81/75.06	  complement(join(top, complement(X)))
74.81/75.06	= { by lemma 29 }
74.81/75.06	  complement(top)
74.81/75.06	= { by lemma 19 }
74.81/75.06	  zero
74.81/75.06	
74.81/75.06	Lemma 35: join(zero, meet(X, X)) = X.
74.81/75.06	Proof:
74.81/75.06	  join(zero, meet(X, X))
74.81/75.06	= { by axiom 8 (def_zero_13) }
74.81/75.06	  join(meet(X, complement(X)), meet(X, X))
74.81/75.06	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.06	  join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
74.81/75.06	= { by lemma 31 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 36: join(X, meet(Y, Y)) = join(Y, meet(X, X)).
74.81/75.06	Proof:
74.81/75.06	  join(X, meet(Y, Y))
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(meet(Y, Y), X)
74.81/75.06	= { by lemma 35 }
74.81/75.06	  join(meet(Y, Y), join(zero, meet(X, X)))
74.81/75.06	= { by axiom 6 (maddux2_join_associativity_2) }
74.81/75.06	  join(join(meet(Y, Y), zero), meet(X, X))
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(join(zero, meet(Y, Y)), meet(X, X))
74.81/75.06	= { by lemma 35 }
74.81/75.06	  join(Y, meet(X, X))
74.81/75.06	
74.81/75.06	Lemma 37: join(X, zero) = X.
74.81/75.06	Proof:
74.81/75.06	  join(X, zero)
74.81/75.06	= { by lemma 34 }
74.81/75.06	  join(X, meet(zero, zero))
74.81/75.06	= { by lemma 36 }
74.81/75.06	  join(zero, meet(X, X))
74.81/75.06	= { by lemma 35 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 38: join(zero, X) = X.
74.81/75.06	Proof:
74.81/75.06	  join(zero, X)
74.81/75.06	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.06	  join(X, zero)
74.81/75.06	= { by lemma 37 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 39: meet(X, top) = X.
74.81/75.06	Proof:
74.81/75.06	  meet(X, top)
74.81/75.06	= { by lemma 38 }
74.81/75.06	  join(zero, meet(X, top))
74.81/75.06	= { by lemma 32 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 40: complement(complement(X)) = X.
74.81/75.06	Proof:
74.81/75.06	  complement(complement(X))
74.81/75.06	= { by lemma 38 }
74.81/75.06	  complement(join(zero, complement(X)))
74.81/75.06	= { by lemma 20 }
74.81/75.06	  meet(X, top)
74.81/75.06	= { by lemma 39 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 41: join(meet(Y, X), meet(X, complement(Y))) = X.
74.81/75.06	Proof:
74.81/75.06	  join(meet(Y, X), meet(X, complement(Y)))
74.81/75.06	= { by lemma 18 }
74.81/75.06	  join(meet(X, Y), meet(X, complement(Y)))
74.81/75.06	= { by lemma 33 }
74.81/75.06	  X
74.81/75.06	
74.81/75.06	Lemma 42: meet(top, X) = X.
74.81/75.06	Proof:
74.81/75.06	  meet(top, X)
74.81/75.06	= { by lemma 18 }
74.81/75.06	  meet(X, top)
74.81/75.06	= { by lemma 39 }
74.81/75.07	  X
74.81/75.07	
74.81/75.07	Lemma 43: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
74.81/75.07	Proof:
74.81/75.07	  complement(join(complement(X), meet(Y, Z)))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  complement(join(complement(X), meet(Z, Y)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  complement(join(meet(Z, Y), complement(X)))
74.81/75.07	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.07	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
74.81/75.07	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.07	  meet(join(complement(Z), complement(Y)), X)
74.81/75.07	= { by lemma 18 }
74.81/75.07	  meet(X, join(complement(Z), complement(Y)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  meet(X, join(complement(Y), complement(Z)))
74.81/75.07	
74.81/75.07	Lemma 44: join(complement(X), complement(Y)) = complement(meet(X, Y)).
74.81/75.07	Proof:
74.81/75.07	  join(complement(X), complement(Y))
74.81/75.07	= { by lemma 42 }
74.81/75.07	  meet(top, join(complement(X), complement(Y)))
74.81/75.07	= { by lemma 43 }
74.81/75.07	  complement(join(complement(top), meet(X, Y)))
74.81/75.07	= { by lemma 19 }
74.81/75.07	  complement(join(zero, meet(X, Y)))
74.81/75.07	= { by lemma 38 }
74.81/75.07	  complement(meet(X, Y))
74.81/75.07	
74.81/75.07	Lemma 45: complement(meet(X, complement(Y))) = join(Y, complement(X)).
74.81/75.07	Proof:
74.81/75.07	  complement(meet(X, complement(Y)))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  complement(meet(complement(Y), X))
74.81/75.07	= { by lemma 38 }
74.81/75.07	  complement(meet(join(zero, complement(Y)), X))
74.81/75.07	= { by lemma 44 }
74.81/75.07	  join(complement(join(zero, complement(Y))), complement(X))
74.81/75.07	= { by lemma 20 }
74.81/75.07	  join(meet(Y, top), complement(X))
74.81/75.07	= { by lemma 39 }
74.81/75.07	  join(Y, complement(X))
74.81/75.07	
74.81/75.07	Lemma 46: meet(X, meet(Y, Z)) = meet(Z, meet(X, Y)).
74.81/75.07	Proof:
74.81/75.07	  meet(X, meet(Y, Z))
74.81/75.07	= { by lemma 39 }
74.81/75.07	  meet(meet(X, meet(Y, Z)), top)
74.81/75.07	= { by lemma 20 }
74.81/75.07	  complement(join(zero, complement(meet(X, meet(Y, Z)))))
74.81/75.07	= { by lemma 44 }
74.81/75.07	  complement(join(zero, join(complement(X), complement(meet(Y, Z)))))
74.81/75.07	= { by lemma 44 }
74.81/75.07	  complement(join(zero, join(complement(X), join(complement(Y), complement(Z)))))
74.81/75.07	= { by axiom 6 (maddux2_join_associativity_2) }
74.81/75.07	  complement(join(zero, join(join(complement(X), complement(Y)), complement(Z))))
74.81/75.07	= { by lemma 45 }
74.81/75.07	  complement(join(zero, complement(meet(Z, complement(join(complement(X), complement(Y)))))))
74.81/75.07	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.07	  complement(join(zero, complement(meet(Z, meet(X, Y)))))
74.81/75.07	= { by lemma 20 }
74.81/75.07	  meet(meet(Z, meet(X, Y)), top)
74.81/75.07	= { by lemma 39 }
74.81/75.07	  meet(Z, meet(X, Y))
74.81/75.07	
74.81/75.07	Lemma 47: meet(meet(Z, X), Y) = meet(X, meet(Y, Z)).
74.81/75.07	Proof:
74.81/75.07	  meet(meet(Z, X), Y)
74.81/75.07	= { by lemma 18 }
74.81/75.07	  meet(Y, meet(Z, X))
74.81/75.07	= { by lemma 46 }
74.81/75.07	  meet(X, meet(Y, Z))
74.81/75.07	
74.81/75.07	Lemma 48: converse(join(X, converse(Y))) = join(Y, converse(X)).
74.81/75.07	Proof:
74.81/75.07	  converse(join(X, converse(Y)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  converse(join(converse(Y), X))
74.81/75.07	= { by axiom 11 (converse_additivity_9) }
74.81/75.07	  join(converse(converse(Y)), converse(X))
74.81/75.07	= { by axiom 5 (converse_idempotence_8) }
74.81/75.07	  join(Y, converse(X))
74.81/75.07	
74.81/75.07	Lemma 49: converse(join(converse(X), Y)) = join(X, converse(Y)).
74.81/75.07	Proof:
74.81/75.07	  converse(join(converse(X), Y))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  converse(join(Y, converse(X)))
74.81/75.07	= { by lemma 48 }
74.81/75.07	  join(X, converse(Y))
74.81/75.07	
74.81/75.07	Lemma 50: join(X, converse(top)) = converse(top).
74.81/75.07	Proof:
74.81/75.07	  join(X, converse(top))
74.81/75.07	= { by lemma 49 }
74.81/75.07	  converse(join(converse(X), top))
74.81/75.07	= { by lemma 30 }
74.81/75.07	  converse(top)
74.81/75.07	
74.81/75.07	Lemma 51: converse(top) = top.
74.81/75.07	Proof:
74.81/75.07	  converse(top)
74.81/75.07	= { by lemma 50 }
74.81/75.07	  join(?, converse(top))
74.81/75.07	= { by lemma 50 }
74.81/75.07	  join(?, join(complement(?), converse(top)))
74.81/75.07	= { by lemma 27 }
74.81/75.07	  join(converse(top), top)
74.81/75.07	= { by lemma 30 }
74.81/75.07	  top
74.81/75.07	
74.81/75.07	Lemma 52: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
74.81/75.07	Proof:
74.81/75.07	  join(meet(X, Y), complement(join(Y, complement(X))))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  join(meet(X, Y), complement(join(complement(X), Y)))
74.81/75.07	= { by lemma 31 }
74.81/75.07	  X
74.81/75.07	
74.81/75.07	Lemma 53: join(meet(Y, complement(X)), complement(join(X, Y))) = complement(X).
74.81/75.07	Proof:
74.81/75.07	  join(meet(Y, complement(X)), complement(join(X, Y)))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  join(meet(complement(X), Y), complement(join(X, Y)))
74.81/75.07	= { by lemma 38 }
74.81/75.07	  join(meet(join(zero, complement(X)), Y), complement(join(X, Y)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  join(meet(join(zero, complement(X)), Y), complement(join(Y, X)))
74.81/75.07	= { by lemma 39 }
74.81/75.07	  join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top))))
74.81/75.07	= { by lemma 20 }
74.81/75.07	  join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X))))))
74.81/75.07	= { by lemma 52 }
74.81/75.07	  join(zero, complement(X))
74.81/75.07	= { by lemma 38 }
74.81/75.07	  complement(X)
74.81/75.07	
74.81/75.07	Lemma 54: meet(complement(X), converse(complement(converse(X)))) = complement(X).
74.81/75.07	Proof:
74.81/75.07	  meet(complement(X), converse(complement(converse(X))))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  meet(converse(complement(converse(X))), complement(X))
74.81/75.07	= { by lemma 37 }
74.81/75.07	  join(meet(converse(complement(converse(X))), complement(X)), zero)
74.81/75.07	= { by lemma 19 }
74.81/75.07	  join(meet(converse(complement(converse(X))), complement(X)), complement(top))
74.81/75.07	= { by lemma 51 }
74.81/75.07	  join(meet(converse(complement(converse(X))), complement(X)), complement(converse(top)))
74.81/75.07	= { by axiom 9 (def_top_12) }
74.81/75.07	  join(meet(converse(complement(converse(X))), complement(X)), complement(converse(join(converse(X), complement(converse(X))))))
74.81/75.07	= { by lemma 49 }
74.81/75.07	  join(meet(converse(complement(converse(X))), complement(X)), complement(join(X, converse(complement(converse(X))))))
74.81/75.07	= { by lemma 53 }
74.81/75.07	  complement(X)
74.81/75.07	
74.81/75.07	Lemma 55: join(complement(converse(X)), converse(join(X, Y))) = top.
74.81/75.07	Proof:
74.81/75.07	  join(complement(converse(X)), converse(join(X, Y)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  join(complement(converse(X)), converse(join(Y, X)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  join(converse(join(Y, X)), complement(converse(X)))
74.81/75.07	= { by axiom 11 (converse_additivity_9) }
74.81/75.07	  join(join(converse(Y), converse(X)), complement(converse(X)))
74.81/75.07	= { by axiom 6 (maddux2_join_associativity_2) }
74.81/75.07	  join(converse(Y), join(converse(X), complement(converse(X))))
74.81/75.07	= { by axiom 9 (def_top_12) }
74.81/75.07	  join(converse(Y), top)
74.81/75.07	= { by lemma 30 }
74.81/75.07	  top
74.81/75.07	
74.81/75.07	Lemma 56: complement(converse(X)) = converse(complement(X)).
74.81/75.07	Proof:
74.81/75.07	  complement(converse(X))
74.81/75.07	= { by axiom 5 (converse_idempotence_8) }
74.81/75.07	  converse(converse(complement(converse(X))))
74.81/75.07	= { by lemma 41 }
74.81/75.07	  converse(join(meet(X, converse(complement(converse(X)))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by axiom 5 (converse_idempotence_8) }
74.81/75.07	  converse(join(meet(converse(converse(X)), converse(complement(converse(X)))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  converse(join(meet(converse(complement(converse(X))), converse(converse(X))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 42 }
74.81/75.07	  converse(join(meet(converse(complement(converse(X))), meet(top, converse(converse(X)))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 47 }
74.81/75.07	  converse(join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), top), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 55 }
74.81/75.07	  converse(join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), converse(join(meet(converse(complement(converse(converse(X)))), complement(converse(X))), complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X)))))))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 31 }
74.81/75.07	  converse(join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), converse(converse(complement(converse(converse(X))))))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  converse(join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(converse(converse(complement(converse(converse(X))))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by axiom 5 (converse_idempotence_8) }
74.81/75.07	  converse(join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(converse(X))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  converse(join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(converse(X))), complement(converse(meet(complement(converse(X)), converse(complement(converse(converse(X))))))))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 54 }
74.81/75.07	  converse(join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(converse(X))), complement(converse(complement(converse(X)))))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 44 }
74.81/75.07	  converse(join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), complement(meet(converse(converse(X)), converse(complement(converse(X)))))), meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by axiom 8 (def_zero_13) }
74.81/75.07	  converse(join(zero, meet(converse(complement(converse(X))), complement(X))))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  converse(join(zero, meet(complement(X), converse(complement(converse(X))))))
74.81/75.07	= { by lemma 38 }
74.81/75.07	  converse(meet(complement(X), converse(complement(converse(X)))))
74.81/75.07	= { by lemma 54 }
74.81/75.07	  converse(complement(X))
74.81/75.07	
74.81/75.07	Lemma 57: join(complement(one), composition(converse(X), complement(X))) = complement(one).
74.81/75.07	Proof:
74.81/75.07	  join(complement(one), composition(converse(X), complement(X)))
74.81/75.07	= { by axiom 3 (composition_identity_6) }
74.81/75.07	  join(complement(one), composition(converse(X), complement(composition(X, one))))
74.81/75.07	= { by lemma 24 }
74.81/75.07	  complement(one)
74.81/75.07	
74.81/75.07	Lemma 58: join(complement(one), converse(complement(one))) = complement(one).
74.81/75.07	Proof:
74.81/75.07	  join(complement(one), converse(complement(one)))
74.81/75.07	= { by axiom 3 (composition_identity_6) }
74.81/75.07	  join(complement(one), composition(converse(complement(one)), one))
74.81/75.07	= { by lemma 38 }
74.81/75.07	  join(complement(one), composition(converse(join(zero, complement(one))), one))
74.81/75.07	= { by lemma 39 }
74.81/75.07	  join(complement(one), composition(converse(join(zero, complement(one))), meet(one, top)))
74.81/75.07	= { by lemma 20 }
74.81/75.07	  join(complement(one), composition(converse(join(zero, complement(one))), complement(join(zero, complement(one)))))
74.81/75.07	= { by lemma 57 }
74.81/75.07	  complement(one)
74.81/75.07	
74.81/75.07	Lemma 59: converse(join(X, complement(one))) = join(converse(X), complement(one)).
74.81/75.07	Proof:
74.81/75.07	  converse(join(X, complement(one)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  converse(join(complement(one), X))
74.81/75.07	= { by axiom 11 (converse_additivity_9) }
74.81/75.07	  join(converse(complement(one)), converse(X))
74.81/75.07	= { by lemma 58 }
74.81/75.07	  join(converse(join(complement(one), converse(complement(one)))), converse(X))
74.81/75.07	= { by lemma 48 }
74.81/75.07	  join(join(complement(one), converse(complement(one))), converse(X))
74.81/75.07	= { by lemma 58 }
74.81/75.07	  join(complement(one), converse(X))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  join(converse(X), complement(one))
74.81/75.07	
74.81/75.07	Lemma 60: join(complement(one), converse(complement(X))) = converse(complement(meet(X, one))).
74.81/75.07	Proof:
74.81/75.07	  join(complement(one), converse(complement(X)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  join(converse(complement(X)), complement(one))
74.81/75.07	= { by lemma 59 }
74.81/75.07	  converse(join(complement(X), complement(one)))
74.81/75.07	= { by lemma 44 }
74.81/75.07	  converse(complement(meet(X, one)))
74.81/75.07	
74.81/75.07	Lemma 61: meet(X, X) = X.
74.81/75.07	Proof:
74.81/75.07	  meet(X, X)
74.81/75.07	= { by lemma 38 }
74.81/75.07	  join(zero, meet(X, X))
74.81/75.07	= { by lemma 35 }
74.81/75.07	  X
74.81/75.07	
74.81/75.07	Lemma 62: complement(join(Y, complement(X))) = meet(X, complement(Y)).
74.81/75.07	Proof:
74.81/75.07	  complement(join(Y, complement(X)))
74.81/75.07	= { by lemma 61 }
74.81/75.07	  complement(join(Y, meet(complement(X), complement(X))))
74.81/75.07	= { by lemma 36 }
74.81/75.07	  complement(join(complement(X), meet(Y, Y)))
74.81/75.07	= { by lemma 43 }
74.81/75.07	  meet(X, join(complement(Y), complement(Y)))
74.81/75.07	= { by lemma 44 }
74.81/75.07	  meet(X, complement(meet(Y, Y)))
74.81/75.07	= { by lemma 61 }
74.81/75.07	  meet(X, complement(Y))
74.81/75.07	
74.81/75.07	Lemma 63: complement(join(complement(X), Y)) = meet(X, complement(Y)).
74.81/75.07	Proof:
74.81/75.07	  complement(join(complement(X), Y))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  complement(join(Y, complement(X)))
74.81/75.07	= { by lemma 62 }
74.81/75.07	  meet(X, complement(Y))
74.81/75.07	
74.81/75.07	Lemma 64: converse(meet(X, one)) = meet(one, converse(X)).
74.81/75.07	Proof:
74.81/75.07	  converse(meet(X, one))
74.81/75.07	= { by lemma 40 }
74.81/75.07	  converse(complement(complement(meet(X, one))))
74.81/75.07	= { by lemma 56 }
74.81/75.07	  complement(converse(complement(meet(X, one))))
74.81/75.07	= { by lemma 60 }
74.81/75.07	  complement(join(complement(one), converse(complement(X))))
74.81/75.07	= { by lemma 63 }
74.81/75.07	  meet(one, complement(converse(complement(X))))
74.81/75.07	= { by lemma 56 }
74.81/75.07	  meet(one, converse(complement(complement(X))))
74.81/75.07	= { by lemma 40 }
74.81/75.07	  meet(one, converse(X))
74.81/75.07	
74.81/75.07	Lemma 65: meet(converse(X), converse(join(X, Y))) = converse(X).
74.81/75.07	Proof:
74.81/75.07	  meet(converse(X), converse(join(X, Y)))
74.81/75.07	= { by lemma 37 }
74.81/75.07	  join(meet(converse(X), converse(join(X, Y))), zero)
74.81/75.07	= { by lemma 19 }
74.81/75.07	  join(meet(converse(X), converse(join(X, Y))), complement(top))
74.81/75.07	= { by lemma 55 }
74.81/75.07	  join(meet(converse(X), converse(join(X, Y))), complement(join(complement(converse(X)), converse(join(X, Y)))))
74.81/75.07	= { by lemma 31 }
74.81/75.07	  converse(X)
74.81/75.07	
74.81/75.07	Lemma 66: meet(join(X, converse(Y)), converse(meet(Y, Z))) = converse(meet(Y, Z)).
74.81/75.07	Proof:
74.81/75.07	  meet(join(X, converse(Y)), converse(meet(Y, Z)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  meet(join(converse(Y), X), converse(meet(Y, Z)))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  meet(join(converse(Y), X), converse(meet(Z, Y)))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  meet(converse(meet(Z, Y)), join(converse(Y), X))
74.81/75.07	= { by lemma 41 }
74.81/75.07	  meet(converse(meet(Z, Y)), join(converse(join(meet(Z, Y), meet(Y, complement(Z)))), X))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  meet(converse(meet(Z, Y)), join(X, converse(join(meet(Z, Y), meet(Y, complement(Z))))))
74.81/75.07	= { by lemma 48 }
74.81/75.07	  meet(converse(meet(Z, Y)), converse(join(join(meet(Z, Y), meet(Y, complement(Z))), converse(X))))
74.81/75.07	= { by axiom 6 (maddux2_join_associativity_2) }
74.81/75.07	  meet(converse(meet(Z, Y)), converse(join(meet(Z, Y), join(meet(Y, complement(Z)), converse(X)))))
74.81/75.07	= { by lemma 65 }
74.81/75.07	  converse(meet(Z, Y))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  converse(meet(Y, Z))
74.81/75.07	
74.81/75.07	Lemma 67: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
74.81/75.07	Proof:
74.81/75.07	  meet(X, meet(Y, Z))
74.81/75.07	= { by lemma 46 }
74.81/75.07	  meet(Y, meet(Z, X))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  meet(Y, meet(X, Z))
74.81/75.07	
74.81/75.07	Lemma 68: meet(X, join(X, Y)) = X.
74.81/75.07	Proof:
74.81/75.07	  meet(X, join(X, Y))
74.81/75.07	= { by axiom 5 (converse_idempotence_8) }
74.81/75.07	  meet(converse(converse(X)), join(X, Y))
74.81/75.07	= { by axiom 5 (converse_idempotence_8) }
74.81/75.07	  meet(converse(converse(X)), converse(converse(join(X, Y))))
74.81/75.07	= { by axiom 11 (converse_additivity_9) }
74.81/75.07	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
74.81/75.07	= { by lemma 65 }
74.81/75.07	  converse(converse(X))
74.81/75.07	= { by axiom 5 (converse_idempotence_8) }
74.81/75.07	  X
74.81/75.07	
74.81/75.07	Lemma 69: meet(X, join(Y, X)) = X.
74.81/75.07	Proof:
74.81/75.07	  meet(X, join(Y, X))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  meet(X, join(X, Y))
74.81/75.07	= { by lemma 68 }
74.81/75.07	  X
74.81/75.07	
74.81/75.07	Lemma 70: join(X, meet(X, complement(Y))) = X.
74.81/75.07	Proof:
74.81/75.07	  join(X, meet(X, complement(Y)))
74.81/75.07	= { by lemma 62 }
74.81/75.07	  join(X, complement(join(Y, complement(X))))
74.81/75.07	= { by lemma 45 }
74.81/75.07	  complement(meet(join(Y, complement(X)), complement(X)))
74.81/75.07	= { by lemma 18 }
74.81/75.07	  complement(meet(complement(X), join(Y, complement(X))))
74.81/75.07	= { by lemma 69 }
74.81/75.07	  complement(complement(X))
74.81/75.07	= { by lemma 40 }
74.81/75.07	  X
74.81/75.07	
74.81/75.07	Lemma 71: meet(complement(X), complement(Y)) = complement(join(X, Y)).
74.81/75.07	Proof:
74.81/75.07	  meet(complement(X), complement(Y))
74.81/75.07	= { by lemma 38 }
74.81/75.07	  meet(join(zero, complement(X)), complement(Y))
74.81/75.07	= { by lemma 62 }
74.81/75.07	  complement(join(Y, complement(join(zero, complement(X)))))
74.81/75.07	= { by lemma 20 }
74.81/75.07	  complement(join(Y, meet(X, top)))
74.81/75.07	= { by lemma 39 }
74.81/75.07	  complement(join(Y, X))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  complement(join(X, Y))
74.81/75.07	
74.81/75.07	Lemma 72: join(X, complement(join(X, Y))) = join(X, complement(Y)).
74.81/75.07	Proof:
74.81/75.07	  join(X, complement(join(X, Y)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  join(complement(join(X, Y)), X)
74.81/75.07	= { by lemma 70 }
74.81/75.07	  join(complement(join(X, join(Y, meet(Y, complement(X))))), X)
74.81/75.07	= { by axiom 6 (maddux2_join_associativity_2) }
74.81/75.07	  join(complement(join(join(X, Y), meet(Y, complement(X)))), X)
74.81/75.07	= { by lemma 71 }
74.81/75.07	  join(meet(complement(join(X, Y)), complement(meet(Y, complement(X)))), X)
74.81/75.07	= { by lemma 40 }
74.81/75.07	  join(meet(complement(join(X, Y)), complement(meet(Y, complement(X)))), complement(complement(X)))
74.81/75.07	= { by lemma 53 }
74.81/75.07	  join(meet(complement(join(X, Y)), complement(meet(Y, complement(X)))), complement(join(meet(Y, complement(X)), complement(join(X, Y)))))
74.81/75.07	= { by lemma 53 }
74.81/75.07	  complement(meet(Y, complement(X)))
74.81/75.07	= { by lemma 45 }
74.81/75.07	  join(X, complement(Y))
74.81/75.07	
74.81/75.07	Lemma 73: meet(X, join(Y, complement(X))) = meet(X, Y).
74.81/75.07	Proof:
74.81/75.07	  meet(X, join(Y, complement(X)))
74.81/75.07	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.07	  meet(X, join(complement(X), Y))
74.81/75.07	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.07	  complement(join(complement(X), complement(join(complement(X), Y))))
74.81/75.07	= { by lemma 72 }
74.81/75.07	  complement(join(complement(X), complement(Y)))
74.81/75.07	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.08	  meet(X, Y)
74.81/75.08	
74.81/75.08	Lemma 74: meet(X, join(complement(X), Y)) = meet(X, Y).
74.81/75.08	Proof:
74.81/75.08	  meet(X, join(complement(X), Y))
74.81/75.08	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.08	  meet(X, join(Y, complement(X)))
74.81/75.08	= { by lemma 73 }
74.81/75.08	  meet(X, Y)
74.81/75.08	
74.81/75.08	Lemma 75: converse(one) = one.
74.81/75.08	Proof:
74.81/75.08	  converse(one)
74.81/75.08	= { by axiom 3 (composition_identity_6) }
74.81/75.08	  composition(converse(one), one)
74.81/75.08	= { by lemma 22 }
74.81/75.08	  one
74.81/75.08	
74.81/75.08	Lemma 76: join(X, complement(meet(X, Y))) = top.
74.81/75.08	Proof:
74.81/75.08	  join(X, complement(meet(X, Y)))
74.81/75.08	= { by lemma 18 }
74.81/75.08	  join(X, complement(meet(Y, X)))
74.81/75.08	= { by lemma 44 }
74.81/75.08	  join(X, join(complement(Y), complement(X)))
74.81/75.08	= { by lemma 28 }
74.81/75.08	  join(complement(Y), top)
74.81/75.08	= { by lemma 30 }
74.81/75.08	  top
74.81/75.08	
74.81/75.08	Lemma 77: join(X, converse(complement(meet(Y, converse(X))))) = top.
74.81/75.08	Proof:
74.81/75.08	  join(X, converse(complement(meet(Y, converse(X)))))
74.81/75.08	= { by lemma 18 }
74.81/75.08	  join(X, converse(complement(meet(converse(X), Y))))
74.81/75.08	= { by lemma 49 }
74.81/75.08	  converse(join(converse(X), complement(meet(converse(X), Y))))
74.81/75.08	= { by lemma 76 }
74.81/75.08	  converse(top)
74.81/75.08	= { by lemma 51 }
74.81/75.08	  top
74.81/75.08	
74.81/75.08	Lemma 78: converse(join(complement(one), X)) = join(converse(X), complement(one)).
74.81/75.08	Proof:
74.81/75.08	  converse(join(complement(one), X))
74.81/75.08	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.08	  converse(join(X, complement(one)))
74.81/75.08	= { by lemma 59 }
74.81/75.08	  join(converse(X), complement(one))
74.81/75.08	
74.81/75.08	Lemma 79: join(X, meet(Y, complement(X))) = join(X, Y).
74.81/75.08	Proof:
74.81/75.08	  join(X, meet(Y, complement(X)))
74.81/75.08	= { by lemma 62 }
74.81/75.08	  join(X, complement(join(X, complement(Y))))
74.81/75.08	= { by lemma 72 }
74.81/75.08	  join(X, complement(complement(Y)))
74.81/75.08	= { by lemma 40 }
74.81/75.08	  join(X, Y)
74.81/75.08	
74.81/75.08	Lemma 80: join(X, converse(meet(Y, converse(complement(X))))) = join(X, converse(Y)).
74.81/75.08	Proof:
74.81/75.08	  join(X, converse(meet(Y, converse(complement(X)))))
74.81/75.08	= { by lemma 56 }
74.81/75.08	  join(X, converse(meet(Y, complement(converse(X)))))
74.81/75.08	= { by lemma 49 }
74.81/75.08	  converse(join(converse(X), meet(Y, complement(converse(X)))))
74.81/75.08	= { by lemma 79 }
74.81/75.08	  converse(join(converse(X), Y))
74.81/75.08	= { by lemma 49 }
74.81/75.08	  join(X, converse(Y))
74.81/75.08	
74.81/75.08	Lemma 81: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
74.81/75.08	Proof:
74.81/75.08	  converse(composition(X, converse(Y)))
74.81/75.08	= { by axiom 10 (converse_multiplicativity_10) }
74.81/75.08	  composition(converse(converse(Y)), converse(X))
74.81/75.08	= { by axiom 5 (converse_idempotence_8) }
74.81/75.08	  composition(Y, converse(X))
74.81/75.08	
74.81/75.08	Lemma 82: join(composition(Y, converse(Z)), converse(composition(Z, X))) = composition(join(Y, converse(X)), converse(Z)).
74.81/75.08	Proof:
74.81/75.08	  join(composition(Y, converse(Z)), converse(composition(Z, X)))
74.81/75.08	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.08	  join(converse(composition(Z, X)), composition(Y, converse(Z)))
74.81/75.08	= { by axiom 10 (converse_multiplicativity_10) }
74.81/75.08	  join(composition(converse(X), converse(Z)), composition(Y, converse(Z)))
74.81/75.08	= { by axiom 13 (composition_distributivity_7) }
74.81/75.08	  composition(join(converse(X), Y), converse(Z))
74.81/75.08	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.08	  composition(join(Y, converse(X)), converse(Z))
74.81/75.08	
74.81/75.08	Lemma 83: join(X, converse(zero)) = X.
74.81/75.08	Proof:
74.81/75.08	  join(X, converse(zero))
74.81/75.08	= { by lemma 49 }
74.81/75.08	  converse(join(converse(X), zero))
74.81/75.08	= { by lemma 37 }
74.81/75.08	  converse(converse(X))
74.81/75.08	= { by axiom 5 (converse_idempotence_8) }
74.81/75.08	  X
74.81/75.08	
74.81/75.08	Lemma 84: converse(zero) = zero.
74.81/75.08	Proof:
74.81/75.08	  converse(zero)
74.81/75.08	= { by lemma 38 }
74.81/75.08	  join(zero, converse(zero))
74.81/75.08	= { by lemma 83 }
74.81/75.08	  zero
74.81/75.08	
74.81/75.08	Lemma 85: composition(join(X, complement(composition(top, Y))), converse(Y)) = composition(X, converse(Y)).
74.81/75.08	Proof:
74.81/75.08	  composition(join(X, complement(composition(top, Y))), converse(Y))
74.81/75.08	= { by axiom 5 (converse_idempotence_8) }
74.81/75.08	  composition(join(X, converse(converse(complement(composition(top, Y))))), converse(Y))
74.81/75.08	= { by lemma 82 }
74.81/75.08	  join(composition(X, converse(Y)), converse(composition(Y, converse(complement(composition(top, Y))))))
74.81/75.08	= { by lemma 38 }
74.81/75.08	  join(composition(X, converse(Y)), converse(join(zero, composition(Y, converse(complement(composition(top, Y)))))))
74.81/75.08	= { by lemma 84 }
74.81/75.08	  join(composition(X, converse(Y)), converse(join(converse(zero), composition(Y, converse(complement(composition(top, Y)))))))
74.81/75.08	= { by lemma 19 }
74.81/75.08	  join(composition(X, converse(Y)), converse(join(converse(complement(top)), composition(Y, converse(complement(composition(top, Y)))))))
74.81/75.08	= { by lemma 56 }
74.81/75.08	  join(composition(X, converse(Y)), converse(join(complement(converse(top)), composition(Y, converse(complement(composition(top, Y)))))))
74.81/75.08	= { by axiom 5 (converse_idempotence_8) }
74.81/75.08	  join(composition(X, converse(Y)), converse(join(complement(converse(top)), composition(converse(converse(Y)), converse(complement(composition(top, Y)))))))
74.81/75.08	= { by lemma 56 }
74.81/75.08	  join(composition(X, converse(Y)), converse(join(complement(converse(top)), composition(converse(converse(Y)), complement(converse(composition(top, Y)))))))
74.81/75.08	= { by axiom 10 (converse_multiplicativity_10) }
74.81/75.08	  join(composition(X, converse(Y)), converse(join(complement(converse(top)), composition(converse(converse(Y)), complement(composition(converse(Y), converse(top)))))))
74.81/75.08	= { by lemma 24 }
74.81/75.08	  join(composition(X, converse(Y)), converse(complement(converse(top))))
74.81/75.08	= { by lemma 56 }
74.81/75.08	  join(composition(X, converse(Y)), converse(converse(complement(top))))
74.81/75.08	= { by lemma 19 }
74.81/75.08	  join(composition(X, converse(Y)), converse(converse(zero)))
74.81/75.08	= { by lemma 84 }
74.81/75.08	  join(composition(X, converse(Y)), converse(zero))
74.81/75.08	= { by lemma 83 }
74.81/75.08	  composition(X, converse(Y))
74.81/75.08	
74.81/75.08	Lemma 86: composition(X, converse(join(Y, complement(composition(top, X))))) = composition(X, converse(Y)).
74.81/75.08	Proof:
74.81/75.08	  composition(X, converse(join(Y, complement(composition(top, X)))))
74.81/75.08	= { by lemma 81 }
74.81/75.08	  converse(composition(join(Y, complement(composition(top, X))), converse(X)))
74.81/75.08	= { by lemma 85 }
74.81/75.08	  converse(composition(Y, converse(X)))
74.81/75.08	= { by lemma 81 }
74.81/75.08	  composition(X, converse(Y))
74.81/75.08	
74.81/75.08	Lemma 87: composition(X, converse(join(complement(composition(top, X)), Y))) = composition(X, converse(Y)).
74.81/75.08	Proof:
74.81/75.08	  composition(X, converse(join(complement(composition(top, X)), Y)))
74.81/75.08	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.08	  composition(X, converse(join(Y, complement(composition(top, X)))))
74.81/75.08	= { by lemma 86 }
74.81/75.09	  composition(X, converse(Y))
74.81/75.09	
74.81/75.09	Lemma 88: composition(meet(X, one), meet(one, converse(X))) = meet(X, one).
74.81/75.09	Proof:
74.81/75.09	  composition(meet(X, one), meet(one, converse(X)))
74.81/75.09	= { by lemma 64 }
74.81/75.09	  composition(meet(X, one), converse(meet(X, one)))
74.81/75.09	= { by lemma 18 }
74.81/75.09	  composition(meet(X, one), converse(meet(one, X)))
74.81/75.09	= { by lemma 73 }
74.81/75.09	  composition(meet(X, one), converse(meet(one, join(X, complement(one)))))
74.81/75.09	= { by axiom 5 (converse_idempotence_8) }
74.81/75.09	  composition(meet(X, one), converse(meet(one, join(converse(converse(X)), complement(one)))))
74.81/75.09	= { by lemma 78 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, converse(join(complement(one), converse(X))))))
74.81/75.10	= { by lemma 80 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, converse(join(complement(one), converse(meet(X, converse(complement(complement(one))))))))))
74.81/75.10	= { by lemma 78 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(converse(converse(meet(X, converse(complement(complement(one)))))), complement(one)))))
74.81/75.10	= { by axiom 5 (converse_idempotence_8) }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(meet(X, converse(complement(complement(one)))), complement(one)))))
74.81/75.10	= { by lemma 40 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(meet(X, converse(one)), complement(one)))))
74.81/75.10	= { by lemma 75 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(meet(X, one), complement(one)))))
74.81/75.10	= { by lemma 45 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, complement(meet(one, complement(meet(X, one)))))))
74.81/75.10	= { by lemma 44 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(complement(one), complement(complement(meet(X, one)))))))
74.81/75.10	= { by lemma 23 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(one, complement(complement(meet(X, one))))))))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), complement(one)))))
74.81/75.10	= { by lemma 57 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), join(complement(one), composition(converse(complement(meet(X, one))), complement(complement(meet(X, one)))))))))
74.81/75.10	= { by lemma 26 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(complement(one), join(composition(converse(complement(meet(X, one))), complement(complement(meet(X, one)))), composition(one, complement(complement(meet(X, one)))))))))
74.81/75.10	= { by axiom 13 (composition_distributivity_7) }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(converse(complement(meet(X, one))), one), complement(complement(meet(X, one))))))))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(one, converse(complement(meet(X, one)))), complement(complement(meet(X, one))))))))
74.81/75.10	= { by lemma 75 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(one, converse(complement(meet(X, converse(one))))), complement(complement(meet(X, one))))))))
74.81/75.10	= { by lemma 77 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(top, complement(complement(meet(X, one))))))))
74.81/75.10	= { by lemma 40 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(top, meet(X, one))))))
74.81/75.10	= { by lemma 74 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, composition(top, meet(X, one)))))
74.81/75.10	= { by lemma 40 }
74.81/75.10	  composition(meet(X, one), converse(meet(one, complement(complement(composition(top, meet(X, one)))))))
74.81/75.10	= { by lemma 87 }
74.81/75.10	  composition(meet(X, one), converse(join(complement(composition(top, meet(X, one))), meet(one, complement(complement(composition(top, meet(X, one))))))))
74.81/75.10	= { by lemma 79 }
74.81/75.10	  composition(meet(X, one), converse(join(complement(composition(top, meet(X, one))), one)))
74.81/75.10	= { by lemma 87 }
74.81/75.10	  composition(meet(X, one), converse(one))
74.81/75.10	= { by lemma 75 }
74.81/75.10	  composition(meet(X, one), one)
74.81/75.10	= { by axiom 3 (composition_identity_6) }
74.81/75.10	  meet(X, one)
74.81/75.10	
74.81/75.10	Lemma 89: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
74.81/75.10	Proof:
74.81/75.10	  join(complement(composition(X, Y)), composition(join(X, Z), Y))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  join(complement(composition(X, Y)), composition(join(Z, X), Y))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  join(composition(join(Z, X), Y), complement(composition(X, Y)))
74.81/75.10	= { by axiom 13 (composition_distributivity_7) }
74.81/75.10	  join(join(composition(Z, Y), composition(X, Y)), complement(composition(X, Y)))
74.81/75.10	= { by axiom 6 (maddux2_join_associativity_2) }
74.81/75.10	  join(composition(Z, Y), join(composition(X, Y), complement(composition(X, Y))))
74.81/75.10	= { by axiom 9 (def_top_12) }
74.81/75.10	  join(composition(Z, Y), top)
74.81/75.10	= { by lemma 30 }
74.81/75.10	  top
74.81/75.10	
74.81/75.10	Lemma 90: join(composition(X, Z), complement(composition(meet(X, Y), Z))) = top.
74.81/75.10	Proof:
74.81/75.10	  join(composition(X, Z), complement(composition(meet(X, Y), Z)))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  join(complement(composition(meet(X, Y), Z)), composition(X, Z))
74.81/75.10	= { by lemma 31 }
74.81/75.10	  join(complement(composition(meet(X, Y), Z)), composition(join(meet(X, Y), complement(join(complement(X), Y))), Z))
74.81/75.10	= { by lemma 89 }
74.81/75.10	  top
74.81/75.10	
74.81/75.10	Lemma 91: join(X, complement(composition(meet(Y, one), X))) = top.
74.81/75.10	Proof:
74.81/75.10	  join(X, complement(composition(meet(Y, one), X)))
74.81/75.10	= { by lemma 22 }
74.81/75.10	  join(composition(converse(one), X), complement(composition(meet(Y, one), X)))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  join(composition(converse(one), X), complement(composition(meet(one, Y), X)))
74.81/75.10	= { by lemma 75 }
74.81/75.10	  join(composition(converse(one), X), complement(composition(meet(converse(one), Y), X)))
74.81/75.10	= { by lemma 90 }
74.81/75.10	  top
74.81/75.10	
74.81/75.10	Lemma 92: meet(Y, composition(meet(X, one), Y)) = composition(meet(X, one), Y).
74.81/75.10	Proof:
74.81/75.10	  meet(Y, composition(meet(X, one), Y))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  meet(composition(meet(X, one), Y), Y)
74.81/75.10	= { by lemma 37 }
74.81/75.10	  join(meet(composition(meet(X, one), Y), Y), zero)
74.81/75.10	= { by lemma 19 }
74.81/75.10	  join(meet(composition(meet(X, one), Y), Y), complement(top))
74.81/75.10	= { by lemma 91 }
74.81/75.10	  join(meet(composition(meet(X, one), Y), Y), complement(join(Y, complement(composition(meet(X, one), Y)))))
74.81/75.10	= { by lemma 52 }
74.81/75.10	  composition(meet(X, one), Y)
74.81/75.10	
74.81/75.10	Lemma 93: meet(X, meet(one, converse(X))) = meet(X, one).
74.81/75.10	Proof:
74.81/75.10	  meet(X, meet(one, converse(X)))
74.81/75.10	= { by lemma 67 }
74.81/75.10	  meet(one, meet(X, converse(X)))
74.81/75.10	= { by lemma 47 }
74.81/75.10	  meet(meet(converse(X), one), X)
74.81/75.10	= { by lemma 18 }
74.81/75.10	  meet(X, meet(converse(X), one))
74.81/75.10	= { by lemma 69 }
74.81/75.10	  meet(X, meet(meet(converse(X), one), join(one, meet(converse(X), one))))
74.81/75.10	= { by lemma 67 }
74.81/75.10	  meet(meet(converse(X), one), meet(X, join(one, meet(converse(X), one))))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  meet(meet(converse(X), one), meet(X, join(one, meet(one, converse(X)))))
74.81/75.10	= { by lemma 39 }
74.81/75.10	  meet(meet(converse(X), one), meet(X, join(one, meet(one, meet(converse(X), top)))))
74.81/75.10	= { by lemma 20 }
74.81/75.10	  meet(meet(converse(X), one), meet(X, join(one, meet(one, complement(join(zero, complement(converse(X))))))))
74.81/75.10	= { by lemma 70 }
74.81/75.10	  meet(meet(converse(X), one), meet(X, one))
74.81/75.10	= { by lemma 47 }
74.81/75.10	  meet(one, meet(meet(X, one), converse(X)))
74.81/75.10	= { by lemma 47 }
74.81/75.10	  meet(one, meet(one, meet(converse(X), X)))
74.81/75.10	= { by lemma 67 }
74.81/75.10	  meet(one, meet(converse(X), meet(one, X)))
74.81/75.10	= { by lemma 67 }
74.81/75.10	  meet(converse(X), meet(one, meet(one, X)))
74.81/75.10	= { by lemma 47 }
74.81/75.10	  meet(converse(X), meet(meet(X, one), one))
74.81/75.10	= { by lemma 47 }
74.81/75.10	  meet(meet(one, converse(X)), meet(X, one))
74.81/75.10	= { by lemma 88 }
74.81/75.10	  meet(meet(one, converse(X)), composition(meet(X, one), meet(one, converse(X))))
74.81/75.10	= { by lemma 92 }
74.81/75.10	  composition(meet(X, one), meet(one, converse(X)))
74.81/75.10	= { by lemma 88 }
74.81/75.10	  meet(X, one)
74.81/75.10	
74.81/75.10	Lemma 94: converse(meet(X, converse(Y))) = meet(Y, converse(X)).
74.81/75.10	Proof:
74.81/75.10	  converse(meet(X, converse(Y)))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  converse(meet(converse(Y), X))
74.81/75.10	= { by lemma 65 }
74.81/75.10	  meet(converse(meet(converse(Y), X)), converse(join(meet(converse(Y), X), complement(join(complement(converse(Y)), X)))))
74.81/75.10	= { by lemma 31 }
74.81/75.10	  meet(converse(meet(converse(Y), X)), converse(converse(Y)))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  meet(converse(converse(Y)), converse(meet(converse(Y), X)))
74.81/75.10	= { by axiom 5 (converse_idempotence_8) }
74.81/75.10	  meet(Y, converse(meet(converse(Y), X)))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  meet(Y, converse(meet(X, converse(Y))))
74.81/75.10	= { by lemma 40 }
74.81/75.10	  meet(Y, converse(meet(X, converse(complement(complement(Y))))))
74.81/75.10	= { by lemma 74 }
74.81/75.10	  meet(Y, join(complement(Y), converse(meet(X, converse(complement(complement(Y)))))))
74.81/75.10	= { by lemma 80 }
74.81/75.10	  meet(Y, join(complement(Y), converse(X)))
74.81/75.10	= { by lemma 74 }
74.81/75.10	  meet(Y, converse(X))
74.81/75.10	
74.81/75.10	Lemma 95: meet(one, converse(X)) = meet(X, one).
74.81/75.10	Proof:
74.81/75.10	  meet(one, converse(X))
74.81/75.10	= { by lemma 64 }
74.81/75.10	  converse(meet(X, one))
74.81/75.10	= { by lemma 66 }
74.81/75.10	  meet(join(?, converse(X)), converse(meet(X, one)))
74.81/75.10	= { by lemma 93 }
74.81/75.10	  meet(join(?, converse(X)), converse(meet(X, meet(one, converse(X)))))
74.81/75.10	= { by lemma 66 }
74.81/75.10	  converse(meet(X, meet(one, converse(X))))
74.81/75.10	= { by lemma 67 }
74.81/75.10	  converse(meet(one, meet(X, converse(X))))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  converse(meet(one, meet(converse(X), X)))
74.81/75.10	= { by lemma 47 }
74.81/75.10	  converse(meet(meet(X, one), converse(X)))
74.81/75.10	= { by lemma 94 }
74.81/75.10	  meet(X, converse(meet(X, one)))
74.81/75.10	= { by lemma 64 }
74.81/75.10	  meet(X, meet(one, converse(X)))
74.81/75.10	= { by lemma 93 }
74.81/75.10	  meet(X, one)
74.81/75.10	
74.81/75.10	Lemma 96: converse(meet(X, one)) = meet(X, one).
74.81/75.10	Proof:
74.81/75.10	  converse(meet(X, one))
74.81/75.10	= { by lemma 40 }
74.81/75.10	  converse(complement(complement(meet(X, one))))
74.81/75.10	= { by lemma 56 }
74.81/75.10	  complement(converse(complement(meet(X, one))))
74.81/75.10	= { by lemma 60 }
74.81/75.10	  complement(join(complement(one), converse(complement(X))))
74.81/75.10	= { by lemma 63 }
74.81/75.10	  meet(one, complement(converse(complement(X))))
74.81/75.10	= { by lemma 56 }
74.81/75.10	  meet(one, converse(complement(complement(X))))
74.81/75.10	= { by lemma 40 }
74.81/75.10	  meet(one, converse(X))
74.81/75.10	= { by lemma 95 }
74.81/75.10	  meet(X, one)
74.81/75.10	
74.81/75.10	Lemma 97: meet(join(Y, X), join(X, complement(Y))) = X.
74.81/75.10	Proof:
74.81/75.10	  meet(join(Y, X), join(X, complement(Y)))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  meet(join(X, Y), join(X, complement(Y)))
74.81/75.10	= { by lemma 45 }
74.81/75.10	  meet(join(X, Y), complement(meet(Y, complement(X))))
74.81/75.10	= { by lemma 62 }
74.81/75.10	  complement(join(meet(Y, complement(X)), complement(join(X, Y))))
74.81/75.10	= { by lemma 53 }
74.81/75.10	  complement(complement(X))
74.81/75.10	= { by lemma 40 }
74.81/75.10	  X
74.81/75.10	
74.81/75.10	Lemma 98: join(complement(Y), meet(Y, X)) = join(X, complement(Y)).
74.81/75.10	Proof:
74.81/75.10	  join(complement(Y), meet(Y, X))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  join(complement(Y), meet(X, Y))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  join(complement(Y), meet(Y, X))
74.81/75.10	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.81/75.10	  join(complement(Y), complement(join(complement(Y), complement(X))))
74.81/75.10	= { by lemma 72 }
74.81/75.10	  join(complement(Y), complement(complement(X)))
74.81/75.10	= { by lemma 44 }
74.81/75.10	  complement(meet(Y, complement(X)))
74.81/75.10	= { by lemma 45 }
74.81/75.10	  join(X, complement(Y))
74.81/75.10	
74.81/75.10	Lemma 99: complement(meet(X, join(complement(Y), Z))) = join(complement(X), meet(Y, complement(Z))).
74.81/75.10	Proof:
74.81/75.10	  complement(meet(X, join(complement(Y), Z)))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  complement(meet(X, join(Z, complement(Y))))
74.81/75.10	= { by lemma 45 }
74.81/75.10	  complement(meet(X, complement(meet(Y, complement(Z)))))
74.81/75.10	= { by lemma 45 }
74.81/75.10	  join(meet(Y, complement(Z)), complement(X))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  join(complement(X), meet(Y, complement(Z)))
74.81/75.10	
74.81/75.10	Lemma 100: composition(meet(X, one), meet(X, one)) = meet(X, one).
74.81/75.10	Proof:
74.81/75.10	  composition(meet(X, one), meet(X, one))
74.81/75.10	= { by lemma 95 }
74.81/75.10	  composition(meet(X, one), meet(one, converse(X)))
74.81/75.10	= { by lemma 88 }
74.81/75.10	  meet(X, one)
74.81/75.10	
74.81/75.10	Lemma 101: composition(converse(X), complement(composition(X, top))) = zero.
74.81/75.10	Proof:
74.81/75.10	  composition(converse(X), complement(composition(X, top)))
74.81/75.10	= { by lemma 38 }
74.81/75.10	  join(zero, composition(converse(X), complement(composition(X, top))))
74.81/75.10	= { by lemma 19 }
74.81/75.10	  join(complement(top), composition(converse(X), complement(composition(X, top))))
74.81/75.10	= { by lemma 24 }
74.81/75.10	  complement(top)
74.81/75.10	= { by lemma 19 }
74.81/75.10	  zero
74.81/75.10	
74.81/75.10	Lemma 102: converse(join(X, composition(converse(Z), Y))) = join(converse(X), composition(converse(Y), Z)).
74.81/75.10	Proof:
74.81/75.10	  converse(join(X, composition(converse(Z), Y)))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  converse(join(composition(converse(Z), Y), X))
74.81/75.10	= { by axiom 11 (converse_additivity_9) }
74.81/75.10	  join(converse(composition(converse(Z), Y)), converse(X))
74.81/75.10	= { by lemma 21 }
74.81/75.10	  join(composition(converse(Y), Z), converse(X))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  join(converse(X), composition(converse(Y), Z))
74.81/75.10	
74.81/75.10	Lemma 103: composition(converse(join(X, complement(composition(Y, top)))), Y) = composition(converse(X), Y).
74.81/75.10	Proof:
74.81/75.10	  composition(converse(join(X, complement(composition(Y, top)))), Y)
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  composition(converse(join(complement(composition(Y, top)), X)), Y)
74.81/75.10	= { by lemma 21 }
74.81/75.10	  converse(composition(converse(Y), join(complement(composition(Y, top)), X)))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  converse(composition(converse(Y), join(X, complement(composition(Y, top)))))
74.81/75.10	= { by axiom 10 (converse_multiplicativity_10) }
74.81/75.10	  composition(converse(join(X, complement(composition(Y, top)))), converse(converse(Y)))
74.81/75.10	= { by axiom 11 (converse_additivity_9) }
74.81/75.10	  composition(join(converse(X), converse(complement(composition(Y, top)))), converse(converse(Y)))
74.81/75.10	= { by lemma 82 }
74.81/75.10	  join(composition(converse(X), converse(converse(Y))), converse(composition(converse(Y), complement(composition(Y, top)))))
74.81/75.10	= { by axiom 10 (converse_multiplicativity_10) }
74.81/75.10	  join(converse(composition(converse(Y), X)), converse(composition(converse(Y), complement(composition(Y, top)))))
74.81/75.10	= { by axiom 11 (converse_additivity_9) }
74.81/75.10	  converse(join(composition(converse(Y), X), composition(converse(Y), complement(composition(Y, top)))))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  converse(join(composition(converse(Y), complement(composition(Y, top))), composition(converse(Y), X)))
74.81/75.10	= { by lemma 101 }
74.81/75.10	  converse(join(zero, composition(converse(Y), X)))
74.81/75.10	= { by lemma 102 }
74.81/75.10	  join(converse(zero), composition(converse(X), Y))
74.81/75.10	= { by lemma 84 }
74.81/75.10	  join(zero, composition(converse(X), Y))
74.81/75.10	= { by lemma 38 }
74.81/75.10	  composition(converse(X), Y)
74.81/75.10	
74.81/75.10	Lemma 104: composition(converse(sk1), complement(sk1)) = zero.
74.81/75.10	Proof:
74.81/75.10	  composition(converse(sk1), complement(sk1))
74.81/75.10	= { by lemma 38 }
74.81/75.10	  join(zero, composition(converse(sk1), complement(sk1)))
74.81/75.10	= { by lemma 19 }
74.81/75.10	  join(complement(top), composition(converse(sk1), complement(sk1)))
74.81/75.10	= { by axiom 17 (goals_17) }
74.81/75.10	  join(complement(top), composition(converse(sk1), complement(composition(sk1, top))))
74.81/75.10	= { by lemma 24 }
74.81/75.10	  complement(top)
74.81/75.10	= { by lemma 19 }
74.81/75.10	  zero
74.81/75.10	
74.81/75.10	Lemma 105: composition(converse(sk1), join(complement(sk1), X)) = composition(converse(sk1), X).
74.81/75.10	Proof:
74.81/75.10	  composition(converse(sk1), join(complement(sk1), X))
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  composition(converse(sk1), join(X, complement(sk1)))
74.81/75.10	= { by axiom 5 (converse_idempotence_8) }
74.81/75.10	  composition(converse(sk1), join(X, converse(converse(complement(sk1)))))
74.81/75.10	= { by lemma 48 }
74.81/75.10	  composition(converse(sk1), converse(join(converse(complement(sk1)), converse(X))))
74.81/75.10	= { by axiom 10 (converse_multiplicativity_10) }
74.81/75.10	  converse(composition(join(converse(complement(sk1)), converse(X)), sk1))
74.81/75.10	= { by axiom 13 (composition_distributivity_7) }
74.81/75.10	  converse(join(composition(converse(complement(sk1)), sk1), composition(converse(X), sk1)))
74.81/75.10	= { by lemma 21 }
74.81/75.10	  converse(join(converse(composition(converse(sk1), complement(sk1))), composition(converse(X), sk1)))
74.81/75.10	= { by lemma 104 }
74.81/75.10	  converse(join(converse(zero), composition(converse(X), sk1)))
74.81/75.10	= { by lemma 84 }
74.81/75.10	  converse(join(zero, composition(converse(X), sk1)))
74.81/75.10	= { by lemma 38 }
74.81/75.10	  converse(composition(converse(X), sk1))
74.81/75.10	= { by lemma 21 }
74.81/75.10	  composition(converse(sk1), X)
74.81/75.10	
74.81/75.10	Lemma 106: composition(top, meet(X, sk1)) = composition(converse(sk1), X).
74.81/75.10	Proof:
74.81/75.10	  composition(top, meet(X, sk1))
74.81/75.10	= { by lemma 51 }
74.81/75.10	  composition(converse(top), meet(X, sk1))
74.81/75.10	= { by lemma 90 }
74.81/75.10	  composition(converse(join(composition(sk1, top), complement(composition(meet(sk1, X), top)))), meet(X, sk1))
74.81/75.10	= { by axiom 17 (goals_17) }
74.81/75.10	  composition(converse(join(sk1, complement(composition(meet(sk1, X), top)))), meet(X, sk1))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  composition(converse(join(sk1, complement(composition(meet(X, sk1), top)))), meet(X, sk1))
74.81/75.10	= { by lemma 103 }
74.81/75.10	  composition(converse(sk1), meet(X, sk1))
74.81/75.10	= { by lemma 40 }
74.81/75.10	  composition(converse(sk1), meet(X, complement(complement(sk1))))
74.81/75.10	= { by lemma 105 }
74.81/75.10	  composition(converse(sk1), join(complement(sk1), meet(X, complement(complement(sk1)))))
74.81/75.10	= { by lemma 79 }
74.81/75.10	  composition(converse(sk1), join(complement(sk1), X))
74.81/75.10	= { by lemma 105 }
74.81/75.10	  composition(converse(sk1), X)
74.81/75.10	
74.81/75.10	Lemma 107: composition(top, meet(sk1, X)) = composition(converse(sk1), X).
74.81/75.10	Proof:
74.81/75.10	  composition(top, meet(sk1, X))
74.81/75.10	= { by lemma 18 }
74.81/75.10	  composition(top, meet(X, sk1))
74.81/75.10	= { by lemma 106 }
74.81/75.10	  composition(converse(sk1), X)
74.81/75.10	
74.81/75.10	Lemma 108: meet(zero, X) = zero.
74.81/75.10	Proof:
74.81/75.10	  meet(zero, X)
74.81/75.10	= { by lemma 18 }
74.81/75.10	  meet(X, zero)
74.81/75.10	= { by lemma 34 }
74.81/75.10	  zero
74.81/75.10	
74.81/75.10	Lemma 109: join(top, X) = top.
74.81/75.10	Proof:
74.81/75.10	  join(top, X)
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  join(X, top)
74.81/75.10	= { by lemma 30 }
74.81/75.10	  top
74.81/75.10	
74.81/75.10	Lemma 110: join(Y, composition(X, Y)) = composition(join(X, one), Y).
74.81/75.10	Proof:
74.81/75.10	  join(Y, composition(X, Y))
74.81/75.10	= { by lemma 23 }
74.81/75.10	  join(composition(one, Y), composition(X, Y))
74.81/75.10	= { by axiom 13 (composition_distributivity_7) }
74.81/75.10	  composition(join(one, X), Y)
74.81/75.10	= { by axiom 1 (maddux1_join_commutativity_1) }
74.81/75.10	  composition(join(X, one), Y)
74.81/75.10	
74.81/75.10	Lemma 111: composition(top, zero) = zero.
74.81/75.10	Proof:
74.81/75.10	  composition(top, zero)
74.81/75.10	= { by lemma 109 }
74.81/75.10	  composition(join(top, one), zero)
74.81/75.10	= { by lemma 51 }
74.81/75.10	  composition(join(converse(top), one), zero)
74.81/75.10	= { by lemma 19 }
74.81/75.10	  composition(join(converse(top), one), complement(top))
74.81/75.10	= { by lemma 110 }
74.81/75.10	  join(complement(top), composition(converse(top), complement(top)))
74.81/75.10	= { by lemma 109 }
74.81/75.10	  join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
74.81/75.10	= { by lemma 110 }
74.81/75.10	  join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
74.81/75.10	= { by lemma 109 }
74.81/75.10	  join(complement(top), composition(converse(top), complement(composition(top, top))))
74.81/75.10	= { by lemma 24 }
74.81/75.10	  complement(top)
74.81/75.10	= { by lemma 19 }
74.81/75.10	  zero
74.81/75.10	
74.81/75.10	Lemma 112: composition(X, zero) = zero.
74.81/75.10	Proof:
74.81/75.10	  composition(X, zero)
74.81/75.10	= { by lemma 38 }
74.81/75.10	  join(zero, composition(X, zero))
74.81/75.10	= { by lemma 111 }
74.81/75.10	  join(composition(top, zero), composition(X, zero))
74.81/75.10	= { by axiom 13 (composition_distributivity_7) }
74.81/75.10	  composition(join(top, X), zero)
74.81/75.10	= { by lemma 109 }
74.81/75.10	  composition(top, zero)
74.81/75.10	= { by lemma 111 }
74.97/75.18	  zero
74.97/75.18	
74.97/75.18	Lemma 113: composition(meet(X, one), composition(meet(X, one), Y)) = composition(meet(X, one), Y).
74.97/75.18	Proof:
74.97/75.18	  composition(meet(X, one), composition(meet(X, one), Y))
74.97/75.18	= { by lemma 95 }
74.97/75.18	  composition(meet(X, one), composition(meet(one, converse(X)), Y))
74.97/75.18	= { by axiom 3 (composition_identity_6) }
74.97/75.18	  composition(meet(X, one), composition(meet(one, composition(converse(X), one)), Y))
74.97/75.18	= { by axiom 12 (composition_associativity_5) }
74.97/75.18	  composition(composition(meet(X, one), meet(one, composition(converse(X), one))), Y)
74.97/75.18	= { by axiom 3 (composition_identity_6) }
74.97/75.18	  composition(composition(meet(X, composition(one, one)), meet(one, composition(converse(X), one))), Y)
74.97/75.18	= { by lemma 75 }
74.97/75.18	  composition(composition(meet(X, composition(one, converse(one))), meet(one, composition(converse(X), one))), Y)
74.97/75.18	= { by axiom 16 (dedekind_law_14) }
74.97/75.18	  composition(join(meet(composition(X, one), one), composition(meet(X, composition(one, converse(one))), meet(one, composition(converse(X), one)))), Y)
74.97/75.18	= { by axiom 3 (composition_identity_6) }
74.97/75.18	  composition(join(meet(X, one), composition(meet(X, composition(one, converse(one))), meet(one, composition(converse(X), one)))), Y)
74.97/75.18	= { by lemma 75 }
74.97/75.18	  composition(join(meet(X, one), composition(meet(X, composition(one, one)), meet(one, composition(converse(X), one)))), Y)
74.97/75.18	= { by axiom 3 (composition_identity_6) }
74.97/75.18	  composition(join(meet(X, one), composition(meet(X, one), meet(one, composition(converse(X), one)))), Y)
74.97/75.18	= { by axiom 5 (converse_idempotence_8) }
74.97/75.18	  composition(converse(converse(join(meet(X, one), composition(meet(X, one), meet(one, composition(converse(X), one)))))), Y)
74.97/75.18	= { by axiom 11 (converse_additivity_9) }
74.97/75.18	  composition(converse(join(converse(meet(X, one)), converse(composition(meet(X, one), meet(one, composition(converse(X), one)))))), Y)
74.97/75.18	= { by lemma 22 }
74.97/75.18	  composition(converse(join(composition(converse(one), converse(meet(X, one))), converse(composition(meet(X, one), meet(one, composition(converse(X), one)))))), Y)
74.97/75.18	= { by lemma 82 }
74.97/75.18	  composition(converse(composition(join(converse(one), converse(meet(one, composition(converse(X), one)))), converse(meet(X, one)))), Y)
74.97/75.18	= { by axiom 11 (converse_additivity_9) }
74.97/75.18	  composition(converse(composition(converse(join(one, meet(one, composition(converse(X), one)))), converse(meet(X, one)))), Y)
74.97/75.18	= { by axiom 10 (converse_multiplicativity_10) }
74.97/75.18	  composition(converse(converse(composition(meet(X, one), join(one, meet(one, composition(converse(X), one)))))), Y)
74.97/75.18	= { by axiom 1 (maddux1_join_commutativity_1) }
74.97/75.18	  composition(converse(converse(composition(meet(X, one), join(meet(one, composition(converse(X), one)), one)))), Y)
74.97/75.18	= { by axiom 5 (converse_idempotence_8) }
74.97/75.18	  composition(composition(meet(X, one), join(meet(one, composition(converse(X), one)), one)), Y)
74.97/75.18	= { by axiom 1 (maddux1_join_commutativity_1) }
74.97/75.18	  composition(composition(meet(X, one), join(one, meet(one, composition(converse(X), one)))), Y)
74.97/75.18	= { by lemma 40 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, composition(converse(X), one)))), Y)
74.97/75.18	= { by lemma 31 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(meet(composition(converse(X), one), complement(composition(converse(X), top))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 37 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(join(meet(composition(converse(X), one), complement(composition(converse(X), top))), zero), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 108 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(join(meet(composition(converse(X), one), complement(composition(converse(X), top))), meet(zero, complement(composition(converse(X), top)))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 112 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(join(meet(composition(converse(X), one), complement(composition(converse(X), top))), meet(composition(converse(X), zero), complement(composition(converse(X), top)))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 34 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(join(meet(composition(converse(X), one), complement(composition(converse(X), top))), meet(composition(converse(X), meet(one, zero)), complement(composition(converse(X), top)))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 101 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(join(meet(composition(converse(X), one), complement(composition(converse(X), top))), meet(composition(converse(X), meet(one, composition(converse(converse(X)), complement(composition(converse(X), top))))), complement(composition(converse(X), top)))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by axiom 15 (modular_law_1_15) }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(meet(composition(converse(X), meet(one, composition(converse(converse(X)), complement(composition(converse(X), top))))), complement(composition(converse(X), top))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 101 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(meet(composition(converse(X), meet(one, zero)), complement(composition(converse(X), top))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 34 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(meet(composition(converse(X), zero), complement(composition(converse(X), top))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 112 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(meet(zero, complement(composition(converse(X), top))), complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 108 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, join(zero, complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top)))))))), Y)
74.97/75.18	= { by lemma 38 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, complement(join(complement(composition(converse(X), one)), complement(composition(converse(X), top))))))), Y)
74.97/75.18	= { by axiom 2 (maddux4_definiton_of_meet_4) }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(one, meet(composition(converse(X), one), composition(converse(X), top))))), Y)
74.97/75.18	= { by lemma 67 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(composition(converse(X), one), meet(one, composition(converse(X), top))))), Y)
74.97/75.18	= { by lemma 40 }
74.97/75.18	  composition(composition(meet(X, one), join(complement(complement(one)), meet(composition(converse(X), one), complement(complement(meet(one, composition(converse(X), top))))))), Y)
74.97/75.18	= { by lemma 99 }
74.97/75.18	  composition(composition(meet(X, one), complement(meet(complement(one), join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top))))))), Y)
74.97/75.18	= { by lemma 37 }
74.97/75.18	  composition(composition(meet(X, one), complement(join(meet(complement(one), join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top))))), zero))), Y)
74.97/75.18	= { by lemma 18 }
74.97/75.18	  composition(composition(meet(X, one), complement(join(meet(join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top)))), complement(one)), zero))), Y)
74.97/75.18	= { by lemma 19 }
74.97/75.18	  composition(composition(meet(X, one), complement(join(meet(join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top)))), complement(one)), complement(top)))), Y)
74.97/75.18	= { by lemma 109 }
74.97/75.18	  composition(composition(meet(X, one), complement(join(meet(join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top)))), complement(one)), complement(join(top, complement(composition(converse(X), one))))))), Y)
74.97/75.18	= { by lemma 76 }
74.97/75.18	  composition(composition(meet(X, one), complement(join(meet(join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top)))), complement(one)), complement(join(join(one, complement(meet(one, composition(converse(X), top)))), complement(composition(converse(X), one))))))), Y)
74.97/75.18	= { by axiom 6 (maddux2_join_associativity_2) }
74.97/75.18	  composition(composition(meet(X, one), complement(join(meet(join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top)))), complement(one)), complement(join(one, join(complement(meet(one, composition(converse(X), top))), complement(composition(converse(X), one)))))))), Y)
74.97/75.18	= { by axiom 1 (maddux1_join_commutativity_1) }
74.97/75.18	  composition(composition(meet(X, one), complement(join(meet(join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top)))), complement(one)), complement(join(one, join(complement(composition(converse(X), one)), complement(meet(one, composition(converse(X), top))))))))), Y)
74.97/75.18	= { by lemma 53 }
74.97/75.18	  composition(composition(meet(X, one), complement(complement(one))), Y)
74.97/75.18	= { by lemma 40 }
74.97/75.18	  composition(composition(meet(X, one), one), Y)
74.97/75.18	= { by axiom 3 (composition_identity_6) }
74.97/75.18	  composition(meet(X, one), Y)
74.97/75.18	
74.97/75.18	Lemma 114: composition(converse(complement(composition(meet(X, one), Y))), meet(X, one)) = composition(converse(complement(Y)), meet(X, one)).
74.97/75.18	Proof:
74.97/75.18	  composition(converse(complement(composition(meet(X, one), Y))), meet(X, one))
74.97/75.18	= { by lemma 92 }
74.97/75.18	  composition(converse(complement(meet(Y, composition(meet(X, one), Y)))), meet(X, one))
74.97/75.18	= { by lemma 44 }
74.97/75.18	  composition(converse(join(complement(Y), complement(composition(meet(X, one), Y)))), meet(X, one))
74.97/75.18	= { by axiom 11 (converse_additivity_9) }
74.97/75.18	  composition(join(converse(complement(Y)), converse(complement(composition(meet(X, one), Y)))), meet(X, one))
74.97/75.18	= { by lemma 96 }
74.97/75.18	  composition(join(converse(complement(Y)), converse(complement(composition(meet(X, one), Y)))), converse(meet(X, one)))
74.97/75.18	= { by lemma 82 }
74.97/75.18	  join(composition(converse(complement(Y)), converse(meet(X, one))), converse(composition(meet(X, one), complement(composition(meet(X, one), Y)))))
74.97/75.18	= { by lemma 113 }
74.97/75.18	  join(composition(converse(complement(Y)), converse(meet(X, one))), converse(composition(meet(X, one), composition(meet(X, one), complement(composition(meet(X, one), Y))))))
74.97/75.18	= { by lemma 82 }
74.97/75.18	  composition(join(converse(complement(Y)), converse(composition(meet(X, one), complement(composition(meet(X, one), Y))))), converse(meet(X, one)))
74.97/75.18	= { by lemma 75 }
74.97/75.18	  composition(join(converse(complement(Y)), converse(composition(meet(X, converse(one)), complement(composition(meet(X, one), Y))))), converse(meet(X, one)))
74.97/75.18	= { by axiom 10 (converse_multiplicativity_10) }
74.97/75.18	  composition(join(converse(complement(Y)), composition(converse(complement(composition(meet(X, one), Y))), converse(meet(X, converse(one))))), converse(meet(X, one)))
74.97/75.18	= { by lemma 94 }
74.97/75.18	  composition(join(converse(complement(Y)), composition(converse(complement(composition(meet(X, one), Y))), meet(one, converse(X)))), converse(meet(X, one)))
74.97/75.18	= { by lemma 95 }
74.97/75.18	  composition(join(converse(complement(Y)), composition(converse(complement(composition(meet(X, one), Y))), meet(X, one))), converse(meet(X, one)))
74.97/75.18	= { by lemma 102 }
74.97/75.18	  composition(converse(join(complement(Y), composition(converse(meet(X, one)), complement(composition(meet(X, one), Y))))), converse(meet(X, one)))
74.97/75.18	= { by lemma 24 }
74.97/75.18	  composition(converse(complement(Y)), converse(meet(X, one)))
74.97/75.18	= { by lemma 96 }
74.97/75.18	  composition(converse(complement(Y)), meet(X, one))
74.97/75.18	
74.97/75.18	Lemma 115: composition(meet(converse(sk1), X), meet(sk1, one)) = composition(X, meet(sk1, one)).
74.97/75.18	Proof:
74.97/75.18	  composition(meet(converse(sk1), X), meet(sk1, one))
74.97/75.18	= { by lemma 18 }
74.97/75.18	  composition(meet(X, converse(sk1)), meet(sk1, one))
74.97/75.18	= { by axiom 3 (composition_identity_6) }
74.97/75.18	  composition(meet(X, composition(converse(sk1), one)), meet(sk1, one))
74.97/75.18	= { by lemma 107 }
74.97/75.18	  composition(meet(X, composition(top, meet(sk1, one))), meet(sk1, one))
74.97/75.18	= { by lemma 40 }
74.97/75.18	  composition(complement(complement(meet(X, composition(top, meet(sk1, one))))), meet(sk1, one))
74.97/75.18	= { by axiom 5 (converse_idempotence_8) }
74.97/75.18	  composition(converse(converse(complement(complement(meet(X, composition(top, meet(sk1, one))))))), meet(sk1, one))
74.97/75.18	= { by lemma 56 }
74.97/75.18	  composition(converse(complement(converse(complement(meet(X, composition(top, meet(sk1, one))))))), meet(sk1, one))
74.97/75.18	= { by lemma 114 }
74.97/75.18	  composition(converse(complement(composition(meet(sk1, one), converse(complement(meet(X, composition(top, meet(sk1, one)))))))), meet(sk1, one))
74.97/75.18	= { by lemma 44 }
74.97/75.18	  composition(converse(complement(composition(meet(sk1, one), converse(join(complement(X), complement(composition(top, meet(sk1, one)))))))), meet(sk1, one))
74.97/75.18	= { by lemma 86 }
74.97/75.18	  composition(converse(complement(composition(meet(sk1, one), converse(complement(X))))), meet(sk1, one))
74.97/75.18	= { by lemma 114 }
74.97/75.18	  composition(converse(complement(converse(complement(X)))), meet(sk1, one))
74.97/75.18	= { by lemma 56 }
74.97/75.18	  composition(converse(converse(complement(complement(X)))), meet(sk1, one))
74.97/75.18	= { by axiom 5 (converse_idempotence_8) }
74.97/75.18	  composition(complement(complement(X)), meet(sk1, one))
74.97/75.18	= { by lemma 40 }
74.97/75.18	  composition(X, meet(sk1, one))
74.97/75.18	
74.97/75.18	Lemma 116: complement(join(Y, join(Z, complement(X)))) = meet(X, complement(join(Y, Z))).
74.97/75.18	Proof:
74.97/75.18	  complement(join(Y, join(Z, complement(X))))
74.97/75.18	= { by axiom 6 (maddux2_join_associativity_2) }
74.97/75.18	  complement(join(join(Y, Z), complement(X)))
74.97/75.18	= { by lemma 62 }
74.97/75.18	  meet(X, complement(join(Y, Z)))
74.97/75.18	
74.97/75.18	Lemma 117: join(meet(Y, complement(X)), complement(join(Y, X))) = complement(X).
74.97/75.18	Proof:
74.97/75.18	  join(meet(Y, complement(X)), complement(join(Y, X)))
74.97/75.18	= { by axiom 1 (maddux1_join_commutativity_1) }
74.97/75.18	  join(meet(Y, complement(X)), complement(join(X, Y)))
74.97/75.18	= { by lemma 53 }
74.97/75.18	  complement(X)
74.97/75.18	
74.97/75.18	Lemma 118: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
74.97/75.18	Proof:
74.97/75.18	  meet(X, meet(join(X, Y), Z))
74.97/75.18	= { by lemma 46 }
74.97/75.18	  meet(join(X, Y), meet(Z, X))
74.97/75.18	= { by lemma 46 }
74.97/75.18	  meet(Z, meet(X, join(X, Y)))
74.97/75.18	= { by lemma 68 }
74.97/75.18	  meet(Z, X)
74.97/75.18	= { by lemma 18 }
74.98/75.18	  meet(X, Z)
74.98/75.18	
74.98/75.18	Lemma 119: meet(join(X, Z), complement(join(X, Y))) = meet(Z, complement(join(X, Y))).
74.98/75.18	Proof:
74.98/75.18	  meet(join(X, Z), complement(join(X, Y)))
74.98/75.18	= { by axiom 1 (maddux1_join_commutativity_1) }
74.98/75.18	  meet(join(Z, X), complement(join(X, Y)))
74.98/75.18	= { by lemma 116 }
74.98/75.18	  complement(join(X, join(Y, complement(join(Z, X)))))
74.98/75.18	= { by lemma 117 }
74.98/75.18	  join(meet(Z, complement(join(X, join(Y, complement(join(Z, X)))))), complement(join(Z, join(X, join(Y, complement(join(Z, X)))))))
74.98/75.18	= { by lemma 116 }
74.98/75.18	  join(meet(Z, meet(join(Z, X), complement(join(X, Y)))), complement(join(Z, join(X, join(Y, complement(join(Z, X)))))))
74.98/75.18	= { by axiom 1 (maddux1_join_commutativity_1) }
74.98/75.18	  join(meet(Z, meet(join(Z, X), complement(join(X, Y)))), complement(join(Z, join(X, join(complement(join(Z, X)), Y)))))
74.98/75.18	= { by axiom 6 (maddux2_join_associativity_2) }
74.98/75.18	  join(meet(Z, meet(join(Z, X), complement(join(X, Y)))), complement(join(Z, join(join(X, complement(join(Z, X))), Y))))
74.98/75.18	= { by axiom 6 (maddux2_join_associativity_2) }
74.98/75.18	  join(meet(Z, meet(join(Z, X), complement(join(X, Y)))), complement(join(join(Z, join(X, complement(join(Z, X)))), Y)))
74.98/75.18	= { by axiom 6 (maddux2_join_associativity_2) }
74.98/75.18	  join(meet(Z, meet(join(Z, X), complement(join(X, Y)))), complement(join(join(join(Z, X), complement(join(Z, X))), Y)))
74.98/75.18	= { by axiom 9 (def_top_12) }
74.98/75.18	  join(meet(Z, meet(join(Z, X), complement(join(X, Y)))), complement(join(top, Y)))
74.98/75.18	= { by lemma 109 }
74.98/75.18	  join(meet(Z, meet(join(Z, X), complement(join(X, Y)))), complement(top))
74.98/75.18	= { by lemma 19 }
74.98/75.18	  join(meet(Z, meet(join(Z, X), complement(join(X, Y)))), zero)
74.98/75.18	= { by lemma 37 }
74.98/75.18	  meet(Z, meet(join(Z, X), complement(join(X, Y))))
74.98/75.18	= { by lemma 118 }
74.98/75.19	  meet(Z, complement(join(X, Y)))
74.98/75.19	
74.98/75.19	Lemma 120: join(meet(X, Y), meet(Y, Z)) = meet(Y, join(X, Z)).
74.98/75.19	Proof:
74.98/75.19	  join(meet(X, Y), meet(Y, Z))
74.98/75.19	= { by lemma 40 }
74.98/75.19	  join(meet(X, Y), complement(complement(meet(Y, Z))))
74.98/75.19	= { by lemma 98 }
74.98/75.19	  join(complement(complement(meet(Y, Z))), meet(complement(meet(Y, Z)), meet(X, Y)))
74.98/75.19	= { by lemma 40 }
74.98/75.19	  join(meet(Y, Z), meet(complement(meet(Y, Z)), meet(X, Y)))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  join(meet(Z, Y), meet(complement(meet(Y, Z)), meet(X, Y)))
74.98/75.19	= { by lemma 47 }
74.98/75.19	  join(meet(Z, Y), meet(meet(Y, complement(meet(Y, Z))), X))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  join(meet(Z, Y), meet(X, meet(Y, complement(meet(Y, Z)))))
74.98/75.19	= { by lemma 63 }
74.98/75.19	  join(meet(Z, Y), meet(X, complement(join(complement(Y), meet(Y, Z)))))
74.98/75.19	= { by lemma 119 }
74.98/75.19	  join(meet(Z, Y), meet(join(complement(Y), X), complement(join(complement(Y), meet(Y, Z)))))
74.98/75.19	= { by lemma 98 }
74.98/75.19	  join(meet(Z, Y), meet(join(complement(Y), X), complement(join(Z, complement(Y)))))
74.98/75.19	= { by axiom 1 (maddux1_join_commutativity_1) }
74.98/75.19	  join(meet(Z, Y), meet(join(complement(Y), X), complement(join(complement(Y), Z))))
74.98/75.19	= { by lemma 119 }
74.98/75.19	  join(meet(Z, Y), meet(X, complement(join(complement(Y), Z))))
74.98/75.19	= { by lemma 63 }
74.98/75.19	  join(meet(Z, Y), meet(X, meet(Y, complement(Z))))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  join(meet(Z, Y), meet(X, meet(complement(Z), Y)))
74.98/75.19	= { by axiom 1 (maddux1_join_commutativity_1) }
74.98/75.19	  join(meet(X, meet(complement(Z), Y)), meet(Z, Y))
74.98/75.19	= { by lemma 118 }
74.98/75.19	  join(meet(X, meet(join(X, Z), meet(complement(Z), Y))), meet(Z, Y))
74.98/75.19	= { by lemma 47 }
74.98/75.19	  join(meet(X, meet(meet(Y, join(X, Z)), complement(Z))), meet(Z, Y))
74.98/75.19	= { by lemma 62 }
74.98/75.19	  join(meet(X, complement(join(Z, complement(meet(Y, join(X, Z)))))), meet(Z, Y))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  join(meet(complement(join(Z, complement(meet(Y, join(X, Z))))), X), meet(Z, Y))
74.98/75.19	= { by lemma 71 }
74.98/75.19	  join(meet(meet(complement(Z), complement(complement(meet(Y, join(X, Z))))), X), meet(Z, Y))
74.98/75.19	= { by lemma 47 }
74.98/75.19	  join(meet(complement(complement(meet(Y, join(X, Z)))), meet(X, complement(Z))), meet(Z, Y))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  join(meet(meet(X, complement(Z)), complement(complement(meet(Y, join(X, Z))))), meet(Z, Y))
74.98/75.19	= { by lemma 40 }
74.98/75.19	  join(meet(meet(X, complement(Z)), complement(complement(meet(Y, join(X, Z))))), complement(complement(meet(Z, Y))))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  join(meet(meet(X, complement(Z)), complement(complement(meet(Y, join(X, Z))))), complement(complement(meet(Y, Z))))
74.98/75.19	= { by lemma 44 }
74.98/75.19	  join(meet(meet(X, complement(Z)), complement(complement(meet(Y, join(X, Z))))), complement(join(complement(Y), complement(Z))))
74.98/75.19	= { by lemma 117 }
74.98/75.19	  join(meet(meet(X, complement(Z)), complement(complement(meet(Y, join(X, Z))))), complement(join(complement(Y), join(meet(X, complement(Z)), complement(join(X, Z))))))
74.98/75.19	= { by lemma 26 }
74.98/75.19	  join(meet(meet(X, complement(Z)), complement(complement(meet(Y, join(X, Z))))), complement(join(meet(X, complement(Z)), join(complement(join(X, Z)), complement(Y)))))
74.98/75.19	= { by lemma 44 }
74.98/75.19	  join(meet(meet(X, complement(Z)), complement(complement(meet(Y, join(X, Z))))), complement(join(meet(X, complement(Z)), complement(meet(join(X, Z), Y)))))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  join(meet(meet(X, complement(Z)), complement(complement(meet(Y, join(X, Z))))), complement(join(meet(X, complement(Z)), complement(meet(Y, join(X, Z))))))
74.98/75.19	= { by lemma 117 }
74.98/75.19	  complement(complement(meet(Y, join(X, Z))))
74.98/75.19	= { by lemma 40 }
74.98/75.19	  meet(Y, join(X, Z))
74.98/75.19	
74.98/75.19	Lemma 121: composition(sk1, composition(top, X)) = composition(sk1, X).
74.98/75.19	Proof:
74.98/75.19	  composition(sk1, composition(top, X))
74.98/75.19	= { by axiom 12 (composition_associativity_5) }
74.98/75.19	  composition(composition(sk1, top), X)
74.98/75.19	= { by axiom 17 (goals_17) }
74.98/75.19	  composition(sk1, X)
74.98/75.19	
74.98/75.19	Lemma 122: composition(sk1, composition(converse(sk1), X)) = composition(sk1, meet(X, sk1)).
74.98/75.19	Proof:
74.98/75.19	  composition(sk1, composition(converse(sk1), X))
74.98/75.19	= { by lemma 106 }
74.98/75.19	  composition(sk1, composition(top, meet(X, sk1)))
74.98/75.19	= { by lemma 121 }
74.98/75.19	  composition(sk1, meet(X, sk1))
74.98/75.19	
74.98/75.19	Lemma 123: meet(sk1, composition(sk1, X)) = composition(sk1, X).
74.98/75.19	Proof:
74.98/75.19	  meet(sk1, composition(sk1, X))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  meet(composition(sk1, X), sk1)
74.98/75.19	= { by lemma 40 }
74.98/75.19	  meet(composition(sk1, X), complement(complement(sk1)))
74.98/75.19	= { by lemma 38 }
74.98/75.19	  join(zero, meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 108 }
74.98/75.19	  join(meet(zero, complement(sk1)), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 112 }
74.98/75.19	  join(meet(composition(sk1, zero), complement(sk1)), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 34 }
74.98/75.19	  join(meet(composition(sk1, meet(X, zero)), complement(sk1)), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 104 }
74.98/75.19	  join(meet(composition(sk1, meet(X, composition(converse(sk1), complement(sk1)))), complement(sk1)), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by axiom 15 (modular_law_1_15) }
74.98/75.19	  join(join(meet(composition(sk1, X), complement(sk1)), meet(composition(sk1, meet(X, composition(converse(sk1), complement(sk1)))), complement(sk1))), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 104 }
74.98/75.19	  join(join(meet(composition(sk1, X), complement(sk1)), meet(composition(sk1, meet(X, zero)), complement(sk1))), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 34 }
74.98/75.19	  join(join(meet(composition(sk1, X), complement(sk1)), meet(composition(sk1, zero), complement(sk1))), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 112 }
74.98/75.19	  join(join(meet(composition(sk1, X), complement(sk1)), meet(zero, complement(sk1))), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 108 }
74.98/75.19	  join(join(meet(composition(sk1, X), complement(sk1)), zero), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 37 }
74.98/75.19	  join(meet(composition(sk1, X), complement(sk1)), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 18 }
74.98/75.19	  join(meet(complement(sk1), composition(sk1, X)), meet(composition(sk1, X), complement(complement(sk1))))
74.98/75.19	= { by lemma 41 }
74.98/75.19	  composition(sk1, X)
74.98/75.19	
74.98/75.19	Lemma 124: converse(composition(top, X)) = composition(converse(X), top).
74.98/75.19	Proof:
74.98/75.19	  converse(composition(top, X))
74.98/75.19	= { by axiom 10 (converse_multiplicativity_10) }
74.98/75.19	  composition(converse(X), converse(top))
74.98/75.19	= { by lemma 51 }
74.98/75.19	  composition(converse(X), top)
74.98/75.19	
74.98/75.19	Lemma 125: converse(composition(X, composition(Y, converse(Z)))) = composition(Z, converse(composition(X, Y))).
74.98/75.19	Proof:
74.98/75.19	  converse(composition(X, composition(Y, converse(Z))))
74.98/75.19	= { by axiom 10 (converse_multiplicativity_10) }
74.98/75.19	  composition(converse(composition(Y, converse(Z))), converse(X))
74.98/75.19	= { by lemma 81 }
74.98/75.19	  composition(composition(Z, converse(Y)), converse(X))
74.98/75.19	= { by axiom 12 (composition_associativity_5) }
74.98/75.19	  composition(Z, composition(converse(Y), converse(X)))
74.98/75.19	= { by axiom 10 (converse_multiplicativity_10) }
74.98/75.19	  composition(Z, converse(composition(X, Y)))
74.98/75.19	
74.98/75.19	Lemma 126: composition(top, composition(X, converse(X))) = composition(top, converse(X)).
74.98/75.19	Proof:
74.98/75.19	  composition(top, composition(X, converse(X)))
74.98/75.19	= { by axiom 12 (composition_associativity_5) }
74.98/75.19	  composition(composition(top, X), converse(X))
74.98/75.19	= { by lemma 85 }
74.98/75.19	  composition(join(composition(top, X), complement(composition(top, X))), converse(X))
74.98/75.19	= { by axiom 9 (def_top_12) }
74.98/75.19	  composition(top, converse(X))
74.98/75.19	
74.98/75.19	Lemma 127: converse(composition(X, top)) = composition(top, converse(X)).
74.98/75.19	Proof:
74.98/75.19	  converse(composition(X, top))
74.98/75.19	= { by axiom 10 (converse_multiplicativity_10) }
74.98/75.19	  composition(converse(top), converse(X))
74.98/75.19	= { by lemma 51 }
74.98/75.19	  composition(top, converse(X))
74.98/75.19	
74.98/75.19	Lemma 128: composition(sk1, composition(X, converse(X))) = composition(sk1, converse(X)).
74.98/75.19	Proof:
74.98/75.19	  composition(sk1, composition(X, converse(X)))
74.98/75.19	= { by lemma 121 }
74.98/75.19	  composition(sk1, composition(top, composition(X, converse(X))))
74.98/75.19	= { by lemma 126 }
74.98/75.19	  composition(sk1, composition(top, converse(X)))
74.98/75.19	= { by lemma 121 }
74.98/75.19	  composition(sk1, converse(X))
74.98/75.19	
74.98/75.19	Lemma 129: composition(converse(X), composition(X, converse(sk1))) = converse(composition(sk1, X)).
74.98/75.19	Proof:
74.98/75.19	  composition(converse(X), composition(X, converse(sk1)))
74.98/75.19	= { by axiom 5 (converse_idempotence_8) }
74.98/75.19	  composition(converse(converse(converse(X))), composition(X, converse(sk1)))
74.98/75.19	= { by lemma 81 }
74.98/75.19	  composition(converse(converse(converse(X))), converse(composition(sk1, converse(X))))
74.98/75.19	= { by axiom 10 (converse_multiplicativity_10) }
74.98/75.19	  converse(composition(composition(sk1, converse(X)), converse(converse(X))))
74.98/75.19	= { by axiom 12 (composition_associativity_5) }
74.98/75.19	  converse(composition(sk1, composition(converse(X), converse(converse(X)))))
74.98/75.19	= { by lemma 128 }
74.98/75.19	  converse(composition(sk1, converse(converse(X))))
74.98/75.19	= { by lemma 81 }
74.98/75.19	  composition(converse(X), converse(sk1))
74.98/75.19	= { by axiom 10 (converse_multiplicativity_10) }
76.86/77.07	  converse(composition(sk1, X))
76.86/77.07	
76.86/77.07	Lemma 130: composition(meet(sk1, one), X) = meet(X, sk1).
76.86/77.07	Proof:
76.86/77.07	  composition(meet(sk1, one), X)
76.86/77.07	= { by lemma 96 }
76.86/77.07	  composition(converse(meet(sk1, one)), X)
76.86/77.07	= { by lemma 21 }
76.86/77.07	  converse(composition(converse(X), meet(sk1, one)))
76.86/77.07	= { by lemma 39 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), top))
76.86/77.07	= { by lemma 51 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), converse(top)))
76.86/77.07	= { by lemma 91 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), converse(join(converse(converse(X)), complement(composition(meet(converse(sk1), one), converse(converse(X))))))))
76.86/77.07	= { by lemma 49 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))))))
76.86/77.07	= { by lemma 97 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), meet(join(composition(converse(X), converse(meet(converse(sk1), one))), converse(complement(composition(meet(converse(sk1), one), converse(converse(X)))))), join(converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))), complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))
76.86/77.07	= { by lemma 81 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), meet(join(converse(composition(meet(converse(sk1), one), converse(converse(X)))), converse(complement(composition(meet(converse(sk1), one), converse(converse(X)))))), join(converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))), complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))
76.86/77.07	= { by axiom 11 (converse_additivity_9) }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), meet(converse(join(composition(meet(converse(sk1), one), converse(converse(X))), complement(composition(meet(converse(sk1), one), converse(converse(X)))))), join(converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))), complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))
76.86/77.07	= { by axiom 9 (def_top_12) }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), meet(converse(top), join(converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))), complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))
76.86/77.07	= { by lemma 51 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), meet(top, join(converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))), complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))
76.86/77.07	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), meet(top, join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))))))))
76.86/77.07	= { by lemma 42 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(composition(meet(converse(sk1), one), converse(converse(X)))))))))
76.86/77.07	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), join(converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))), complement(composition(converse(X), converse(meet(converse(sk1), one))))))))
76.86/77.07	= { by lemma 98 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), meet(composition(converse(X), converse(meet(converse(sk1), one))), converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))))))))
76.86/77.07	= { by lemma 40 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), join(complement(complement(complement(composition(converse(X), converse(meet(converse(sk1), one)))))), meet(composition(converse(X), converse(meet(converse(sk1), one))), converse(complement(composition(meet(converse(sk1), one), converse(converse(X))))))))))
76.86/77.07	= { by lemma 56 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), join(complement(complement(complement(composition(converse(X), converse(meet(converse(sk1), one)))))), meet(composition(converse(X), converse(meet(converse(sk1), one))), complement(converse(composition(meet(converse(sk1), one), converse(converse(X))))))))))
76.86/77.07	= { by lemma 99 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))), join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(composition(meet(converse(sk1), one), converse(converse(X))))))))))
76.86/77.07	= { by lemma 18 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(composition(meet(converse(sk1), one), converse(converse(X))))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))
76.86/77.07	= { by lemma 79 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), meet(converse(composition(meet(converse(sk1), one), converse(converse(X)))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))
76.86/77.07	= { by lemma 18 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), meet(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))), converse(composition(meet(converse(sk1), one), converse(converse(X)))))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))
76.86/77.07	= { by lemma 94 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))
76.86/77.07	= { by lemma 40 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))
76.86/77.07	= { by lemma 56 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), complement(converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))
76.86/77.07	= { by lemma 31 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), complement(converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))), complement(join(meet(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))), complement(join(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))), converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))))))))))
76.86/77.07	= { by lemma 77 }
76.86/77.07	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), complement(converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))), complement(join(meet(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))), complement(top))))))))
76.86/77.07	= { by lemma 19 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), complement(converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))), complement(join(meet(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))), zero)))))))
76.86/77.08	= { by lemma 37 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), complement(converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))), complement(meet(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))))))))
76.86/77.08	= { by lemma 62 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(complement(join(meet(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))), complement(join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), complement(converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))))))))))
76.86/77.08	= { by lemma 62 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(complement(join(meet(complement(composition(converse(X), converse(meet(converse(sk1), one)))), converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one)))))))))), meet(converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))))
76.86/77.08	= { by lemma 41 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(complement(converse(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))))))
76.86/77.08	= { by lemma 56 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(converse(complement(complement(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))))))
76.86/77.08	= { by lemma 40 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(converse(meet(composition(meet(converse(sk1), one), converse(converse(X))), converse(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))))
76.86/77.08	= { by lemma 94 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))), converse(composition(meet(converse(sk1), one), converse(converse(X)))))))))
76.86/77.08	= { by lemma 18 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(converse(composition(meet(converse(sk1), one), converse(converse(X)))), complement(complement(composition(converse(X), converse(meet(converse(sk1), one))))))))))
76.86/77.08	= { by lemma 45 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), join(complement(composition(converse(X), converse(meet(converse(sk1), one)))), complement(converse(composition(meet(converse(sk1), one), converse(converse(X)))))))))
76.86/77.08	= { by lemma 44 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(composition(converse(X), converse(meet(converse(sk1), one))), converse(composition(meet(converse(sk1), one), converse(converse(X)))))))))
76.86/77.08	= { by lemma 81 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(meet(composition(converse(X), converse(meet(converse(sk1), one))), composition(converse(X), converse(meet(converse(sk1), one))))))))
76.86/77.08	= { by lemma 61 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(composition(converse(X), converse(meet(converse(sk1), one)))))))
76.86/77.08	= { by lemma 64 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(composition(converse(X), meet(one, converse(converse(sk1))))))))
76.86/77.08	= { by axiom 5 (converse_idempotence_8) }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(composition(converse(X), meet(one, sk1))))))
76.86/77.08	= { by lemma 18 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), join(converse(X), complement(composition(converse(X), meet(sk1, one))))))
76.86/77.08	= { by lemma 73 }
76.86/77.08	  converse(meet(composition(converse(X), meet(sk1, one)), converse(X)))
76.86/77.08	= { by lemma 18 }
76.86/77.08	  converse(meet(converse(X), composition(converse(X), meet(sk1, one))))
76.86/77.08	= { by lemma 100 }
76.86/77.08	  converse(meet(converse(X), composition(converse(X), composition(meet(sk1, one), meet(sk1, one)))))
76.86/77.08	= { by lemma 96 }
76.86/77.08	  converse(meet(converse(X), composition(converse(X), composition(converse(meet(sk1, one)), meet(sk1, one)))))
76.86/77.08	= { by axiom 12 (composition_associativity_5) }
76.86/77.08	  converse(meet(converse(X), composition(composition(converse(X), converse(meet(sk1, one))), meet(sk1, one))))
76.86/77.08	= { by lemma 115 }
76.86/77.08	  converse(meet(converse(X), composition(meet(converse(sk1), composition(converse(X), converse(meet(sk1, one)))), meet(sk1, one))))
76.86/77.08	= { by lemma 18 }
76.86/77.08	  converse(meet(composition(meet(converse(sk1), composition(converse(X), converse(meet(sk1, one)))), meet(sk1, one)), converse(X)))
76.86/77.08	= { by axiom 14 (modular_law_2_16) }
76.86/77.08	  converse(join(meet(composition(converse(sk1), meet(sk1, one)), converse(X)), meet(composition(meet(converse(sk1), composition(converse(X), converse(meet(sk1, one)))), meet(sk1, one)), converse(X))))
76.86/77.08	= { by lemma 18 }
76.86/77.08	  converse(join(meet(composition(converse(sk1), meet(sk1, one)), converse(X)), meet(converse(X), composition(meet(converse(sk1), composition(converse(X), converse(meet(sk1, one)))), meet(sk1, one)))))
76.86/77.08	= { by lemma 115 }
76.86/77.08	  converse(join(meet(composition(converse(sk1), meet(sk1, one)), converse(X)), meet(converse(X), composition(composition(converse(X), converse(meet(sk1, one))), meet(sk1, one)))))
76.86/77.08	= { by lemma 120 }
76.86/77.08	  converse(meet(converse(X), join(composition(converse(sk1), meet(sk1, one)), composition(composition(converse(X), converse(meet(sk1, one))), meet(sk1, one)))))
76.86/77.08	= { by axiom 13 (composition_distributivity_7) }
76.86/77.08	  converse(meet(converse(X), composition(join(converse(sk1), composition(converse(X), converse(meet(sk1, one)))), meet(sk1, one))))
76.86/77.08	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(sk1)), meet(sk1, one))))
76.86/77.08	= { by axiom 17 (goals_17) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(composition(sk1, top))), meet(sk1, one))))
76.86/77.08	= { by lemma 123 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(meet(sk1, composition(sk1, top)))), meet(sk1, one))))
76.86/77.08	= { by axiom 5 (converse_idempotence_8) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(meet(sk1, composition(converse(converse(sk1)), top)))), meet(sk1, one))))
76.86/77.08	= { by lemma 124 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(meet(sk1, converse(composition(top, converse(sk1)))))), meet(sk1, one))))
76.86/77.08	= { by lemma 126 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(meet(sk1, converse(composition(top, composition(sk1, converse(sk1))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 125 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(meet(sk1, composition(sk1, converse(composition(top, sk1)))))), meet(sk1, one))))
76.86/77.08	= { by lemma 124 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(meet(sk1, composition(sk1, composition(converse(sk1), top))))), meet(sk1, one))))
76.86/77.08	= { by lemma 123 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(composition(sk1, composition(converse(sk1), top)))), meet(sk1, one))))
76.86/77.08	= { by lemma 122 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(composition(sk1, meet(top, sk1)))), meet(sk1, one))))
76.86/77.08	= { by lemma 42 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(composition(sk1, sk1))), meet(sk1, one))))
76.86/77.08	= { by axiom 10 (converse_multiplicativity_10) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(converse(sk1), converse(sk1))), meet(sk1, one))))
76.86/77.08	= { by lemma 22 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(composition(converse(one), converse(sk1)), converse(sk1))), meet(sk1, one))))
76.86/77.08	= { by lemma 81 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(converse(composition(sk1, converse(converse(one)))), converse(sk1))), meet(sk1, one))))
76.86/77.08	= { by lemma 121 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(converse(composition(sk1, composition(top, converse(converse(one))))), converse(sk1))), meet(sk1, one))))
76.86/77.08	= { by lemma 125 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(composition(converse(one), converse(composition(sk1, top))), converse(sk1))), meet(sk1, one))))
76.86/77.08	= { by lemma 127 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(composition(converse(one), composition(top, converse(sk1))), converse(sk1))), meet(sk1, one))))
76.86/77.08	= { by lemma 22 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(composition(top, converse(sk1)), converse(sk1))), meet(sk1, one))))
76.86/77.08	= { by axiom 12 (composition_associativity_5) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, composition(converse(sk1), converse(sk1)))), meet(sk1, one))))
76.86/77.08	= { by lemma 81 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(composition(sk1, converse(converse(sk1)))))), meet(sk1, one))))
76.86/77.08	= { by lemma 128 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(composition(sk1, composition(converse(sk1), converse(converse(sk1))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 125 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, composition(converse(sk1), converse(composition(sk1, converse(sk1)))))), meet(sk1, one))))
76.86/77.08	= { by lemma 129 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, composition(converse(sk1), composition(converse(converse(sk1)), composition(converse(sk1), converse(sk1)))))), meet(sk1, one))))
76.86/77.08	= { by axiom 12 (composition_associativity_5) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, composition(composition(converse(sk1), converse(converse(sk1))), composition(converse(sk1), converse(sk1))))), meet(sk1, one))))
76.86/77.08	= { by axiom 12 (composition_associativity_5) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(composition(top, composition(converse(sk1), converse(converse(sk1)))), composition(converse(sk1), converse(sk1)))), meet(sk1, one))))
76.86/77.08	= { by lemma 126 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(composition(top, converse(converse(sk1))), composition(converse(sk1), converse(sk1)))), meet(sk1, one))))
76.86/77.08	= { by axiom 12 (composition_associativity_5) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, composition(converse(converse(sk1)), composition(converse(sk1), converse(sk1))))), meet(sk1, one))))
76.86/77.08	= { by lemma 129 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(composition(sk1, converse(sk1))))), meet(sk1, one))))
76.86/77.08	= { by lemma 97 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(composition(sk1, converse(sk1)), complement(meet(sk1, one))))))), meet(sk1, one))))
76.86/77.08	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(complement(meet(sk1, one)), composition(sk1, converse(sk1))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 100 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(complement(composition(meet(sk1, one), meet(sk1, one))), composition(sk1, converse(sk1))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 96 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(complement(composition(meet(sk1, one), converse(meet(sk1, one)))), composition(sk1, converse(sk1))))))), meet(sk1, one))))
76.86/77.08	= { by axiom 3 (composition_identity_6) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(complement(composition(meet(sk1, one), converse(meet(sk1, one)))), composition(sk1, composition(converse(sk1), one))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 122 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(complement(composition(meet(sk1, one), converse(meet(sk1, one)))), composition(sk1, meet(one, sk1))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 18 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(complement(composition(meet(sk1, one), converse(meet(sk1, one)))), composition(sk1, meet(sk1, one))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 96 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(complement(composition(meet(sk1, one), converse(meet(sk1, one)))), composition(sk1, converse(meet(sk1, one)))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 128 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(complement(composition(meet(sk1, one), converse(meet(sk1, one)))), composition(sk1, composition(meet(sk1, one), converse(meet(sk1, one))))))))), meet(sk1, one))))
76.86/77.08	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(composition(sk1, composition(meet(sk1, one), converse(meet(sk1, one)))), complement(composition(meet(sk1, one), converse(meet(sk1, one))))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 113 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), join(composition(sk1, composition(meet(sk1, one), converse(meet(sk1, one)))), complement(composition(meet(sk1, one), composition(meet(sk1, one), converse(meet(sk1, one)))))))))), meet(sk1, one))))
76.86/77.08	= { by lemma 90 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(meet(join(meet(sk1, one), composition(sk1, converse(sk1))), top)))), meet(sk1, one))))
76.86/77.08	= { by lemma 39 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(join(meet(sk1, one), composition(sk1, converse(sk1)))))), meet(sk1, one))))
76.86/77.08	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), composition(top, converse(join(composition(sk1, converse(sk1)), meet(sk1, one))))), meet(sk1, one))))
76.86/77.08	= { by lemma 127 }
76.86/77.08	  converse(meet(converse(X), composition(join(composition(converse(X), converse(meet(sk1, one))), converse(composition(join(composition(sk1, converse(sk1)), meet(sk1, one)), top))), meet(sk1, one))))
76.86/77.08	= { by lemma 49 }
76.86/77.08	  converse(meet(converse(X), composition(converse(join(converse(composition(converse(X), converse(meet(sk1, one)))), composition(join(composition(sk1, converse(sk1)), meet(sk1, one)), top))), meet(sk1, one))))
76.86/77.08	= { by lemma 103 }
76.86/77.08	  converse(meet(converse(X), composition(converse(join(join(converse(composition(converse(X), converse(meet(sk1, one)))), composition(join(composition(sk1, converse(sk1)), meet(sk1, one)), top)), complement(composition(meet(sk1, one), top)))), meet(sk1, one))))
76.86/77.08	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.08	  converse(meet(converse(X), composition(converse(join(complement(composition(meet(sk1, one), top)), join(converse(composition(converse(X), converse(meet(sk1, one)))), composition(join(composition(sk1, converse(sk1)), meet(sk1, one)), top)))), meet(sk1, one))))
76.86/77.08	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.08	  converse(meet(converse(X), composition(converse(join(complement(composition(meet(sk1, one), top)), join(composition(join(composition(sk1, converse(sk1)), meet(sk1, one)), top), converse(composition(converse(X), converse(meet(sk1, one))))))), meet(sk1, one))))
76.86/77.08	= { by axiom 6 (maddux2_join_associativity_2) }
76.86/77.08	  converse(meet(converse(X), composition(converse(join(join(complement(composition(meet(sk1, one), top)), composition(join(composition(sk1, converse(sk1)), meet(sk1, one)), top)), converse(composition(converse(X), converse(meet(sk1, one)))))), meet(sk1, one))))
76.86/77.08	= { by axiom 1 (maddux1_join_commutativity_1) }
76.86/77.08	  converse(meet(converse(X), composition(converse(join(join(complement(composition(meet(sk1, one), top)), composition(join(meet(sk1, one), composition(sk1, converse(sk1))), top)), converse(composition(converse(X), converse(meet(sk1, one)))))), meet(sk1, one))))
76.86/77.08	= { by lemma 89 }
76.86/77.08	  converse(meet(converse(X), composition(converse(join(top, converse(composition(converse(X), converse(meet(sk1, one)))))), meet(sk1, one))))
76.86/77.08	= { by lemma 109 }
76.86/77.08	  converse(meet(converse(X), composition(converse(top), meet(sk1, one))))
76.86/77.08	= { by lemma 51 }
76.86/77.08	  converse(meet(converse(X), composition(top, meet(sk1, one))))
76.86/77.08	= { by lemma 107 }
76.86/77.08	  converse(meet(converse(X), composition(converse(sk1), one)))
76.86/77.08	= { by axiom 3 (composition_identity_6) }
76.86/77.08	  converse(meet(converse(X), converse(sk1)))
76.86/77.08	= { by lemma 18 }
76.86/77.08	  converse(meet(converse(sk1), converse(X)))
76.86/77.08	= { by lemma 94 }
76.86/77.08	  meet(X, converse(converse(sk1)))
76.86/77.08	= { by axiom 5 (converse_idempotence_8) }
76.86/77.08	  meet(X, sk1)
76.86/77.08	
76.86/77.08	Goal 1 (goals_18): composition(meet(sk1, one), sk2) = join(meet(sk1, sk2), composition(meet(sk1, one), sk2)).
76.86/77.08	Proof:
76.86/77.08	  composition(meet(sk1, one), sk2)
76.86/77.08	= { by lemma 130 }
76.86/77.08	  meet(sk2, sk1)
76.86/77.08	= { by lemma 40 }
76.86/77.08	  meet(sk2, complement(complement(sk1)))
76.86/77.08	= { by lemma 25 }
76.86/77.08	  meet(sk2, join(complement(complement(sk1)), complement(complement(sk1))))
76.86/77.08	= { by lemma 40 }
76.86/77.08	  meet(sk2, join(sk1, complement(complement(sk1))))
76.86/77.08	= { by lemma 40 }
76.86/77.08	  meet(sk2, join(sk1, sk1))
76.86/77.08	= { by lemma 120 }
76.86/77.08	  join(meet(sk1, sk2), meet(sk2, sk1))
76.86/77.08	= { by lemma 130 }
76.86/77.08	  join(meet(sk1, sk2), composition(meet(sk1, one), sk2))
76.86/77.08	% SZS output end Proof
76.86/77.08	
76.86/77.08	RESULT: Unsatisfiable (the axioms are contradictory).
76.86/77.09	EOF